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Bureau of Mines Information Circular/1980 







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t'-(vi 



Bureau of Mines Statistical Projection 
Methodology of U. S. Mineral 
Consumption by End Use: Aluminum 
as an Example 

By William Y. Mo and Barry W. Klein 




UNITED STATES DEPARTMENT OF THE INTERIOR 



'' . . ^ 

Information Circular 8825 

Bureau of Mines Statistical Projection 
Methodology of U. S. Mineral 
Consumption by End Use: Aluminum 
as an Example 

By William Y. Mo and Barry W. Klein 




UNITED STATES DEPARTMENT OF THE INTERIOR 
Cecil D. Andrus, Secretary 

BUREAU OF MINES 

Lindsay D. Norman, Acting Director 






This publication has been cataloged as follows: 



Mo, William Y 

Bureau of Mines statistical projection methodology of U.S. 
mineral consumption by end use: aluminum as an example. 

(Information circular * U.S. Bureau of Mines ; 8825) 
Supt. of Docs, no.: I 28.27:8825. 

1. Mineral industries— United States. 2. Economic forecasting — 
United States— Methodology. 1. Klein, Barry W., joint author. II. Title. 
III. Series: United States. Bureau of Mines. Information circular ; 8825. 



TN295.U4 [HD9506.U62] 622s [338.2'0973] 79-607929 



For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 



CONTENTS 

Page 

Abstract 1 

Introduction 1 

General method 2 

Statistical demand pro j ection procedure 4 

Compilation of data 4 

Selection of independent variables 5 

Selection of estimated equation used for projection 5 

Determination of end-use pro j ection 5 

Aluminum as an example of the statistical demand projection procedure... 6 

Compilation of data 6 

Selection of independent variable 6 

Selection of estimated equation used for projection 6 

Determination of end-use pro j ection 9 

Conclusion 26 

Appendix A. — Macroeconomic variables used in Bureau of Mines statistical 

demand pro j ection system 27 

Appendix B. — Least-squares method, best unbiased estimator, and maximum 

likelihood estimator 28 

Appendix C. — Aluminum consumption data used to derive statistical 

projections 36 

Appendix D. — Data for macroeconomic variables used to derive statistical 

projections 37 

Appendix E. — Macroeconomic explanatory variables used in estimated 

aluminum end-use consumption equations and associated R^ values 41 

Appendix F. — Other computer graphs with lower R^ values 43 

ILLUSTRATIONS 

1. Scatter diagram of aluminum demand for construction use and new 

construction activity • 10 

2. Scatter diagram of aluminum demand for transportation use and 

FRB index for motor vehicles 11 

3. Scatter diagram of aluminum demand for electrical use and FRB index 

for major electrical equipment and parts 12 

4. Scatter diagram of aluminum demand for cans and containers use and 

FRB index for metal cans 13 

5 . Scatter diagram of aluminum demand for appliances and equipment use 

and gross private domestic investment 14 

6. Scatter diagram of aluminum demand for machinery use and FRB index 

for nonelectrical machinery 15 

7. Scatter diagram of aluminum demand for dissipative use and gross 

private domestic investment 16 

8. Scatter diagram of aluminum demand for other metal use and FRB 

index for chemicals and products 17 

9. Scatter diagram of aluminum demand for refractories use and FRB 

index for stone, clay, and glass 18 

10. Scatter diagram of aluminum demand for chemicals use and FRB index 

for paper and products 19 



11 



ILLUSTRATIONS—Continued 

Page 

11. Scatter diagram of aluminum demand for abrasives use and FRB 

index for total production 20 

12 . Aluminum demand for construction use 22 

13 . Aluminum demand for transportation use 22 

14 . Aluminum demand for elec tr ical use 22 

15 . Aluminum demand for cans and containers use 23 

16. Aluminum demand for appliances and equipment use 23 

17 . Aluminum demand for machinery use 23 

18 . Aluminum demand for dissipative use 24 

19 . Aluminum demand for other metal use 24 

20 . Aluminum demand for refractories use 24 

21 . Aluminum demand for chemicals use 25 

22 . Aluminum demand for abrasives use 25 

F-1. Scatter diagram of aluminum demand for construction use and 

gross national product (GNP) 43 

F-2 . Scatter diagram of aluminum demand for construction use and 

U.S. population 44 

F-3. Scatter diagram of aluminum demand for construction use and 

FRB index for total production 45 

F-4. Scatter diagram of aluminum demand for construction use and 

FRB index for construction and allied equipment 46 

F-5 . Scatter diagram of aluminum demand for construction use and 

FRB index for hardware, plumbing, structural metal 47 

F-6. Scatter diagram of aluminum demand for transportation use and 

gross national product (GNP) 48 

F-7. Scatter diagram of aluminum demand for transportation use and 

FRB index for total production 49 

F-8. Scatter diagram of aluminum demand for transportation use and 

FRB index for tires 50 

F-9. Scatter diagram of aluminum demand for transportation use and 

FRB index for automobiles 51 

F-10. Scatter diagram of aluminum demand for transportation use and 

FRB index for transportation equipment 52 

F-11. Scatter diagram of aluminum demand for transportation use and 

FRB index for petroleum products 53 

F-12. Scatter diagram of alimiinum demand for transportation use and 

FRB index for trucks , buses , and trailers 54 

F-13. Scatter diagram of aluminum demand for transportation use and 

FRB index for aircraft and parts 55 

F-14. Scatter diagram of aluminum demand for transportation use and 

FRB index for ships and boats 56 

F-15. Scatter diagram of aluminum demand for electrical use and 

gross national product (GNP) 57 

F-16. Scatter diagram of aluminum demand for electrical use and 

FRB index for total production 58 

F-17. Scatter diagram of aluminum demand for electrical use and 

FRB index for communication equipment 59 



1X1 



ILLUSTRATIONS— Continued 

Page 
F-18. Scatter diagram of aluminum demand for electrical use and 

FRB index for electrical machinery 60 

F-19. Scatter diagram of aluminum demand for electrical use and 

FRB index for household appliances 61 

F-20. Scatter diagram of aluminum demand for electrical use and 

FRB index for fabricated metal products 62 

F-21 . Scatter diagram of aluminum demand for cans and containers use 

and gross national product (GNP) 63 

F-22. Scatter diagram of aluminum demand for cans and containers use 

and FRB index for total production 64 

F-23. Scatter diagram of aluminum demand for cans and containers use 

and FRB index for metalworking machinery 65 

F-24 . Scatter diagram of aluminum demand for cans and containers use 

and FRB index for fabricated metal products 66 

F-25. Scatter diagram of aluminum demand for appliances and equipment 

use and gross national product (GNP) 67 

F-26. Scatter diagram of aluminum demand for appliances and equipment 

use and U.S. population 68 

F-27. Scatter diagram of aluminum demand for appliances and equipment 

use and FRB index for total production 69 

F-28. Scatter diagram of aluminum demand for appliances and equipment 

use and FRB index for major electrical equipment and parts 70 

F-29. Scatter diagram of aluminum demand for appliances and equipment 

use and FRB index for electrical machinery 71 

F-30. Scatter diagram of aluminum demand for appliances and equipment 

use and FRB index for household appliances 72 

F-31. Scatter diagram of aluminum demand for appliances and equipment 

use and FRB index for communication equipment 73 

F-32. Scatter diagram of aluminum demand for machinery use and 

gross national product (GNP) 74 

F-33. Scatter diagram of aluminum demand for machinery use and 

U.S. population 75 

F-34. Scatter diagram of aluminum demand for machinery use and 

FRB index for total production 76 

F-35 . Scatter diagram of aluminum demand for machinery use and 

FRB index for metalworking machinery 77 

F-36 . Scatter diagram of aluminum demand for machinery use and 

FRB index for special and general industrial equipment 78 

F-37 . Scatter diagram of aluminum demand for dissipative use and 

gross national product (GNP) 79 

F-38. Scatter diagram of aluminum demand for dissipative use and 

U.S. population 80 

F-39. Scatter diagram of aluminum demand for dissipative use and 

FRB index for total production 81 

F-40. Scatter diagram of aluminum demand for dissipative use and 

FRB index for chemicals and products 82 

F-41. Scatter diagram of aluminum demand for dissipative use and 

FRB index for basic chemicals 83 

F-42. Scatter diagram of aluminum demand for other metal use and 

gross national product (GNP) - 84 



IV 



ILLUSTRATIONS—Continued 

Page 
F-43. Scatter diagram of aliraiinum demand for other metal use and 

U.S. population 85 

F-44. Scatter diagram of aluminum demand for other metal use and 

gross private domestic investment 86 

F-45. Scatter diagram of aluminum demand for other metal use and 

FRB index for total production . , 87 

F-46. Scatter diagram of aluminum demand for other metal use and 

FRB index for paints 88 

F-47. Scatter diagram of aluminum demand for other metal use and 

FRB index for basic chemicals 89 

F-48 . Scatter diagram of aluminum demand for refractories use and 

gross national product (GNP) 90 

F-49. Scatter diagram of aluminum demand for refractories use and 

U.S. population 91 

F-50. Scatter diagram of aluminum demand for refractories use and 

gross private domestic investment 92 

F-51. Scatter diagram of aluminum demand for refractories use and 

FRB index for total production 93 

F-52. Scatter diagram of aluminum demand for refractories use and 

FRB index for iron and steel 94 

F-53. Scatter diagram of aluminum demand for refractories use and 

FRB index for basic steel and mill products 95 

F-54. Scatter diagram of aluminum demand for chemicals use and 

gross national product (GNP) 96 

F-55 . Scatter diagram of aluminum demand for chemicals use and 

U.S. population 97 

F-56. Scatter diagram of aluminum demand for chemicals use and 

gross private domestic investment 98 

F-57. Scatter diagram of aluminum demand for chemicals use and 

FRB index for total production 99 

F-58. Scatter diagram of aluminum demand for chemicals use and 

FRB index for basic chemicals 100 

F-59. Scatter diagram of aluminum demand for chemicals use and 

FRB index for chemicals and products 101 

F-60. Scatter diagram of aluminum demand for chemicals use and 

FRB index for plastics products, n.e.c 102 

F-61. Scatter diagram of aluminum demand for chemicals use and 

FRB index for textile mill products 103 

F-62. Scatter diagram of aluminum demand for abrasives use and 

gross national product (GNP) 104 

F-63. Scatter diagram of aluminum demand for abrasives use and 

U.S. population 105 

F-64. Scatter diagram of aluminum demand for abrasives use and 

gross private domestic investment 106 

F-65. Scatter diagram of aluminum demand for abrasives use and 

FRB index for iron and steel 107 

F-66. Scatter diagram of aluminum demand for abrasives use and 

FRB index for metalworking machinery 108 

F-67. Scatter diagram of aluminum demand for abrasives use and 

FRB index for basic steel and mill products 109 



V 



TABLES 

Page 

A-1. Macroeconomic variables used in Bureau of Mines statistical demand 

projection system 27 

C-1. Aluminum consumption used to derive statistical projections 36 

D-1. Data for macroeconomic variables used to derive statistical 

proj ections 37 

E-1. Macroeconomic explanatory variables used in estimated aluminum 

end-use consumption equations and associated R values 41 



BUREAU OF MINES STATISTICAL PROJECTION METHODOLOGY 
OF U.S. MINERAL CONSUMPTION BY END USE: ALUMINUM 

AS AN EXAMPLE 

by 

William Y. Mo ^ and Barry W. Klein ^ 



ABSTRACT 

This Information Circular provides a detailed background documentation 
of how the Bureau of Mines projects U.S. mineral demand to the year 2000. 
These statistical projections serve as a quantitative basis for contingency 
forecasting of the low, high, and probable U.S. demand for nonfuel minerals 
by end-use categories and are published in the Bureau of Mines Mineral Com- 
modity Profiles and Mineral Facts and Problems series. Aluminum is used as 
an example in this study. 

INTRODUCTION 

The Bureau of Mines forecasting system consists of two important com- 
ponents: (1) statistical analysis and (2) contingency analysis. The system 
is currently used in deriving long-range U.S. demand forecasts for mineral 
commodities by end-use categories to the year 2000. The statistical analysis 
is performed by the analytical staff of the Division of Analytic Studies, and 
the contingency analysis is carried out by commodity specialists of the 
Division of Production/Consumption Data Collection and Interpretation. The 
statistical analyses are based on a standard least-squares procedure, while 
the contingency analyses are of a qualitative nature based on the commodity 
specialists' knowledge and judgment. The results of statistical analysis 
are referred to as "statistical projections," and the results of contingency 
analysis are called "contingency forecasts." These demand projections and 
forecasts by end use are published in the Bureau of Mines Mineral Commodity 
Profiles and Mineral Facts and Problems series to provide the general public 
and decisionmakers with an overview of present and probable future supply- 
demand relationships for individual mineral industries. 

Although the statistical analysis method used in the Bureau of Mines 
statistical demand projection system is the ordinary least-squares, or regres- 
sion analysis, method, it is not familiar to many users of Bureau demand projec- 
tions and forecasts. Accordingly, this Information Circular provides docu- 
mentation for the methodology and procedure for deriving these projections, 
which are made for all nonfuel minerals, using aluminum as an example. The 

Economist, Branch of Economic Analysis, Bureau of Mines, Washington, D.C. 



main part of the publication is simplified to the extent possible, and a more 
rigorous mathematical derivation is presented in an appendix. 

GENERAL METHOD 

Changes are taking place constantly in a modern society. Decisionmakers, 
in both private and government sectors, not only are faced with present prob- 
lems but also are required to anticipate future events. As a result, projec- 
tion and forecasting activities have become an indispensable part of an 
informed decisionmaking process. Of course, anticipating future occurrences 
has always been a very difficult task. To deal with this problem, many 
statistical methods, both simple and complicated, have been developed and 
applied. Among all the methods that have been developed, regression analysis 
is one of the most frequently used techniques. The Bureau of Mines uses this 
simple technique to derive a set of statistical mineral demand projections by 
end use to the year 2000. 

In the Bureau of Mines statistical demand projection system, it is 
assumed that the end-use consumption of a mineral commodity can be approxi- 
mated by a simple linear regression equation as follows: 



Y. 



a + b X^ + e^ (for t = 1,2,3, .... ,n) (1) 



where Y^ = end-use consumption of a mineral commodity at time t, Xj = a 
macroeconomic variable at time t, e^ = a disturbance term, and a and b are 
parameters. 

Equation 1 specifies that end-use consumption of a mineral commodity is 
related primarily with a macroeconomic variable. Because mineral commodities 
are basic raw materials used in our economy, it is not unreasonable to expect 
that the demand for a mineral commodity will depend upon general economic con- 
ditions, and that the demand for the mineral will change as economic condi- 
tions change. Furthermore, mineral commodities may be used by different 
sectors of the economy in a variety of ways. Therefore, 38 macroeconomic 
variables are selected as possible explanatory variables. A list of these 
variables is given in appendix A. 

Equation 1 is defined basically by two parameters, a and b, but the 
relationship is inexact because it also contains a disturbance (or error) 
term e^ . The inclusion of an error term in equation 1 is to take into account 
the influence of omitted variables, the error of approximation of the func- 
tional form, and o'ther unpredictable random effects. 

The unknown parameters, a and b, must be estimated by reference to 
observable data for some particular historical period. The mathematical 
derivations of the estimation formulas are quite complex (appendix B) . 
Basically, the method attempts to find numerical estimates for the parameters 
a and b that, when used in equation 1, best explain the known historical data. 
In other words, the objective of the method is to estimate parameter values 
that make the accumulated squares of the sample-period prediction errors as 
small as possible. This is commonly known as the least-squares criterion. 



A A 
Let a and b represent the least-squares estimated values of a and b. Then, as 

indicated in appendix B, the a and b can be obtained by using the following 

formulas: 

An — n 2 -2 

b = [E X(Y^- nXY]/[Z X^ - n X ] (2) 

• t=l t=l • 

A _ A_ 

a = Y - bX (3) 

_ n _ n 
where X = (1/n) Z X^ , and Y = (1/n) E Y^ . 

t=l t=l 

Furthermore, with additional assumptions about the disturbance term e^ 
in equation 1, the above least-squares estimators have additional desirable 
properties such as the "best linear unbiased" and "maximum likelihood" fea- 
tures. The detailed explanations are given in appendix B. 

After several estimated equations have been obtained for an end-use con- 
sumption o£ a mineral commodity, it becomes necessary to determine which esti- 
mated equation should be selected as a basis for deriving the statistical 
projection. Obviously, among the estimated equations, the equation that has 
the smallest sum of the squared residuals (or predicted errors) should be 
selected. From the following definition of coefficient of determination 
(commonly known as R^) , it is clear that this is equivalent to selecting the 
estimated equation that has the highest R^ value. 

n A 2 n - 2 
R^ = 1-[E e^ /E (Y^-Y) ] (4) 

t=l t=l 

n A^ n A n A A 
where E e^ = E [Y^ -Y^ ] ^ = E [Y^ - a-b X^ ] ^ 

t=l t=l t=l 

= sum of squared residuals. 

From equation 4 it is clear that the maximum value of R^ must be unity, 

^ Ao 
and this can only occur when E e'^ = 0; that is, when all historical data 

t=l 

points lie exactly on the estimated regression line; the observed values of the 

A 
dependent variable Y would be equal to the predicted values Y calculated from 

the estimated regression line. As the independent (or explanatory) variable 
explains less and less of the variation in the dependent variable, the value 
of R^ falls closer and closer toward zero. Hence, the R^ value provides a 



useful measure that can be used to determine which macroeconomic variable best 
describes the historical pattern of each end-use consumption and, therefore, 
which macroeconomic variable should be used as an explanatory factor for 
deriving each end-use statistical projection. Furthermore, if we assume that 
the relationship shown by the selected estimated equation will continue into 
the future, then projections can be easily obtained by solving this estimated 
equation for the variable to be projected by substituting the appropriate 
period's value for the macroeconomic explanatory variable in the estimated 
equation. 

STATISTICAL DEMAND PROJECTION PROCEDURE 

In the preceding section, the general method used in the Bureau of Mines 
statistical demand projection system was described briefly. The procedures 
for actual implementation of this methodology are explained in this section. 

To increase the implementation efficiency, the Branch of Economic 
Analysis developed a computerized statistical demand projection system. 
However, it is worth bearing in mind that a computer does not "make projec- 
tions," but rather carries out the tedious routine calculations that serve 
as the basis for deriving the projections. This computerized system performs 
statistical regression analyses based on the historical data of each mineral 
commodity's end-use consumption, and produces graphs of the historical rela- 
tionships between U.S. mineral commodity end-use consumption and various 
macroeconomic explanatory variables. It also estimates the underlying regres- 
sion equations and the associated coefficients of determination. These 
results provide information as to which macroeconomic variable best explains 
the historical pattern of end-use consumption, and consequently which esti- 
mated relationship should be used as a basis for deriving end-use consumption 
projections. The implementation of this system involves the following opera- 
tional procedures. 

Compilation of Data 

The first step in this procedure is to compile the dependent-variable 
data, which are annual time series data of end-use consumptions. The recent 
estimated equations were based on data from 1960 through 1977. Portions of 
these data are published in the Bureau of Mines publication "Minerals in the 
U.S. Economy: Ten-Year Supply-Demand Profiles for Nonfuel Mineral Commodities 
(1968-77)." In some cases, the estimated equations were based on, for example, 
only 13 and 14 years of data, because the end-use consumption data were not 
available as far back as 1960. In order to improve the usefulness of the data 
collected, the Bureau of Mines has recently developed canvasses for selected 
commodities, requesting end-use data according to the Standard Industrial 
Classification (SIC) breakdown. 

The U.S. annual data for the independent variables, shown in appendix A, 
are compiled from various Government sources. The U.S. annual data on 
gross national product (GNP), population, gross private domestic investment 
(GPDI) , and new construction activity are all from the Department of Commerce, 
and industrial production indexes are obtained from the Federal Reserve 



Board (FRB) . The data from 1960 through 1977 for these explanatory variables 
are stored on a computer tape for easy retrieval. 

Selection of Independent Variables 

Although the Bureau of Mines presently maintains 38 macroeconomic vari- 
ables as possible explanatory variables, only 4 of them (GNP, population, 
GPDI, and the FRB total industrial production index) are currently used for 
estimating end-use consumption equations for every commodity. In the earlier 
analyses, before improvements were made in the procedure, not all four explan- 
atory variables were used in some cases . These variables are now used in 
every case because they are more general economic indicators than the remain- 
ing industry-specific variables. To determine which of the remaining explan- 
atory variables are logically related to any given end use of a commodity, the 
appropriate commodity specialist is consulted. Therefore, there are a minimum 
of four, and usually more, estimated linear equations for any given end use of 
a commodity. 

Selection of Estimated Equation Used for Projection 

To select the estimated equation used for deriving an end-use projection 
for any given commodity, the R criterion is used, as explained in the 
"General Method" section. To provide users with a clear idea of the goodness 
of fit of the underlying estimated equations, a decision has been made that a 
statistical projection derived from an estimated equation with an R value 
greater than or equal to 0.70 should be distinguished from one with an R 
value below 0.70. This distinction has been indicated in the projections and 
forecasts table in each Mineral Commodity Profile since early 1978 and will 
be made in the 1980 edition of Mineral Facts and Problems. 

Determination of End-Use Projection 

To use the estimated regression equation for making projections, the 
year 2000 value of the explanatory variable in each regression equation has 
to be estimated or assumed. Values of these macroeconomic variables were 
obtained from several sources. The U.S. population value in year 2000 was 
based on a Department of Commerce, Bureau of the Census series. A set of 
consistent estimated values for U.S. GNP and FRB industrial production indexes, 
from 1976 through 1990, was provided by Data" Resources, Inc. (DRI)^ from its 
macroeconomic model. Based on this information, the values for year 2000 
were estimated. From these assumed values and the estimated relationships, 
the end-use projections were obtained by the technique described in the 
"General Method" section. The Bureau is currently having these estimated 
values updated. 

It should be noted that in a few cases a particular end use for a com- 
modity has a rapidly declining historical pattern. Consequently, the projec- 
tion will become zero in, or even before, the year 2000. In these cases we 
assume the year 2000 end-use consumption will be zero. 



^DRI, Lexington, Mass., is one of several consulting firms that provide 
macroeconomic forecasting models to clients on a subscriber fee basis. 



ALUMINUM AS AN EXAMPLE OF THE STATISTICAL DEMAND PROJECTION PROCEDURE 

The demand projection procedure described in the previous section, will 
be illustrated by using aluminum as an example in this section. Aluminum is 
chosen for the illustration because it is one of the important mineral com- 
modities and has a large variety of end-use applications. Also, it was one 
of the commodities used in our earlier prototype studies for testing the 
method . 

Compilation of Data 

Aluminum consumption was disaggregated into 11 end uses. The annual data 
for each end use are shown in appendix C. At the time the Bureau's statisti- 
cal projections were made, the available data for the eight metal end uses 
covered the period 1960-76, and the data for the nonmetal end uses were avail- 
able only for 1966-76. Recently, the data have been revised. Therefore, 
there are small differences between these original data and the newer data, 
published in the "Minerals in the U.S. Economy: Ten-Year Supply-Demand 
Profiles for Nonfuel Mineral Commodities (1968-77)," which are available 
for 1968 to 1977. 

The annual data of the U.S. macroeconomic explanatory variables used in 
estimating the ordinary least-square equations of aluminum end-use consumption 
are shown in appendix D. These data are obtained from various Government 
sources as indicated in the previous section. 

Selection of Independent Variable 

The selection of the macroeconomic variable that was used in each esti- 
mated equation was done in consultation with the aluminum specialist. The 
number of explanatory variables chosen varied from as many as 10 for the 
transportation end use to as few as 5 for the cans and containers end use, 
as shown in appendix E. 

Selection of Estimated Equation Used for Projection 

As shown in appendix E, the explanatory variable with the highest R 
value in each end use is indicated by an asterisk. For example, in the con- 
struction end-use category, the results of the R value range from 0.72 to 
0.95, and the explanatory variable with the highest R^ value is U.S. new 
construction activity. This means that 95 percent of the variation (about 
the mean) in the consumption of aluminum used in construction has been 
explained by the linear influence of U.S. new construction activity. There- 
fore, among the six estimated equations, the following estimated equation with 
U.S. new construction activity as the explanatory variable was selected for 
deriving the statistical demand projection for aluminum used in construction: 



Construction (data period 1960-76) 

Y^ =^1,301.0015 + 20. 9926X^ (5) 

R^ = 0.9549 

where Y^ = aluminum used in construction at time 

t, thousand short tons, 

and Xt = U.S. new construction activity at time 

t, billion 1973 dollars. 

For other end uses, the following estimated equations were selected: 

Transportation (data period 1960-76) 

Y^ =-171.8997+ 7.5565Xt (6) 

r2 = 0.9272 

where Y^ = aluminum used in transportation at time 

t, thousand short tons, 

and Xt = FRB industrial production index for motor 

vehicles at time t, 1967 = 100. 

Electrical (data period 1960-76) 

Y^ = -211.9089 + 7.7377Xt (7) 

R^ = 0.9562 

where Y^ = aluminum for electrical use at time t, 

thousand short tons, 

and Xj = FRB industrial production index for major 

electrical equipment and parts at time t, 
1967 = 100. 

Cans and containers (data period 1960-76) 

Yt = -943.3190 + 13.9370Xt (8) 

R^ = 0.9473 

where Y^ = aluminum used in cans and containers at time t, 

thousand short tons, 

and X( = FRB industrial production index for metal cans 

at time t, 1967 = 100. 



8 

Appliances and equipment (data period 1960-76) 

Y^ = -145.9117 + 3.3436Xt (9) 

r2 = 0.9647 

where Y^ = aluminum used in appliances and equipment at 

time t, thousand short tons, 

and Xj = gross private domestic investment at time t, 

billion 1973 dollars. 

Machinery (data period 1960-76) 

Y^ = -10.8506 + 3.0758X^ (10) 

r2 = 0.8986 

where Y^ = aluminum used in machinery at time t, 

thousand short tons, 

and Xj = FRB industrial production index for nonelectri- 

cal machinery at time t, 1967 = 100. 

Dissipative (data period 1960-76) 

Y^ = -131.2343 + 2.6701Xt (11) 

r2 = 0.7827 

where Y^ = aluminum for dissipative use at time t, 

thousand short tons, 

and Xj = gross private domestic investment at time t, 

billion 1973 dollars. 

Other metal uses (data period 1960-76) 

Y^ = 176.3138 - 0.6843Xt (12) 

r2 = 0.1932 

where Y^ = aluminum for other metal uses at time t, 

thousand short tons, 

and 5Q = FRB industrial production index for chemicals 

and products at time t, 1967 = 100. 



Refractories (data period 1966-76) 

Y^ = -165.1793 + 2.9170Xt (13) 

r2 = 0.7288 

where Y^ = aluminum used in refractories at time t, 

thousand short tons, 

and Xt = FRB industrial production index for stone, 

clay, and glass at time t, 1967 = 100. 

Chemicals (data period 1966-76) 

Yt = -336.7366 + 5.2409Xt (14) 

R^ = 0.8248 

where Y^ = aluminum used in chemicals at time t, 

thousand short tons, 

and Xt = FRB industrial production index for paper 

and products at time t, 1967 = 100. 

Abrasives (data period 1966-76) 

Y^ = -86.8939 + 1.5614Xt (15) 

R^ = 0.6393 

^^^^^ ^t = aluminum used in abrasives at time t, 

thousand short tons, 

and Xj = FRB total industrial production index at 

time t, 1967 = 100. 

Determination of End-Use Projection 

After the appropriate estimated equation has been selected, the end-use 
projection can be obtained by solving the estimated equation for the depend- 
ent variable by substituting the appropriate period's value for the macro- 
economic explanatory variable in the estimated equation. For example, the 
projection of aluminum consumption in the construction end use for year 2000 
is 5,375,000 short tons, which is calculated from estimated equation 5 under 
the assumption that the value of U.S. new construction activity in year 2000 
will be $318 billion; that is — 

Y2000 = -1.301.0015 + 20.9926X^,,„ 

= -1,301.0015 + (20.9926) (318) 

= -1,301.0015 + 6,675.6468 

= 5,374.6 thousand short tons 

By using the same procedure, the other end-use projections can also 
be obtained. 

Figures 1 through 11 are the computer graphs corresponding to the esti- 
mated equations 5 through 15 for the 11 end uses. Sixty-seven similar graphs 
with smaller R^'s appear in appendix F. 



10 



ALUMINUM DEMANDS-CONSTRUCTION 

THOUSAND SHORT TONS < 1960-1976) 



Y 
2000 



1500 - 



1000 - 



500 - 












50 



100 



150 



200 X 



NEW CONSTRUCTION ACTIVITY 
BILLION 1973 $ 



Y = - 1301.0015 + 
R-SQUARED = 0.954-9 



20.9926 X 



PROJECTIONS 
1985 
2000 



2981.5008 
5374.. 8639 



X 
204- . 0000 
318.0000 



FIGURE 1. - Scatter diagram of aluminum demand for construction use and new 
construction activity. 



11 



ALUMINUM DEMANDS-TRANSPORTATION 

THOUSAND SHORT TONS (1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 












50 



100 



FRBI-MOTOR VEHICLES 
1967 = 100 



150 



200 X 



Y = - 171.8997 + 
R-SQUARED = 0.9272 



7.5565 X 



PROJECTIONS 
1985 
2000 



1201.874-3 
2033.0907 



181.8000 
291.8000 



FIGURE 2. - Scatter diagram of aluminum demand for transportation use and FRB index 
for motor vehicles. 



12 



ALUMINUM DEMAND-~£LECTRICAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 

1000 



750 - 



500 - 



250 - 












50 



100 



150 



200 X 



FRBI-MAJOR ELECTRICAL EQUIPMENT & PARTS 
1967 = 100 



Y = ~ 211.9089 + 
R- SQUARED = 0.9562 



7.7377 X 



PROJECTIONS 
1985 
2000 



Y 
1129.0498 
1560.04.46 



173.3000 
229 . 0000 



FIGURE 3. - Scatter diagram of aluminum demand for electrical use and FRB index for 
major electrical equipment and parts. 



13 



ALUMINUM DEMAND — CANS AND CONTAINERS 
THOUSAND SHORT TONS (1960-1976) 



Y 
1200 



900 - 



800 - 



300 - 












50 



100 



150 



200 X 



FRBI -METAL CANS 
1967 = 100 



Y = - 94-3.3190 + 
R-SQUARED = 0.94-73 



13.9370 X 



PROJECTIONS Y X 

1985 14-14.. 8263 169.2000 
2000 194-1 . 64-60 207 . 0000 

FIGURE 4. - Scatter diagram of aluminum demand for cans and containers use and FRB 
index for metal cans. 



14 



ALUMINUM DEMAND — APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS (1980-1976) 



Y 



UJiOU^ 




1 




1 


— r 




750 


- 








/ 


/ 


500 


- 










- 


250 


- 


/ 


/ 






- 


r?i 




y L 




L 


1 









150 



225 



300 X 



GROSS PRIVATE DOMESTIC INVESTMENT 
BILLION 1973 % 



Y = - 145.9117 + 
R-SQUARED = 0.964.7 



3.3436 X 



PROJECTIONS 
1985 
2000 



903.9932 
1592.7843 



314.0000 
520 . 0000 



FIGURE 5. - Scatter diagram of aluminum demand for appliances and equipment use and 
gross private domestic investment. 



15 



ALUMINUM DEMANDS-MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



Y 
600 



450 - 



300 - 



150 












50 



100 



150 



200 X 



FRBI-NONELECTRICAL MACHINERY 
1967 = 100 



Y = -> 10.8506 + 
R -SQUARED = 0.8986 



3.0758 X 



PROJECTIONS 
1985 
2000 



Y 
606.7728 
979.5614 



200.8000 
322 . 0000 



FIGURE 6. - Scatter diagram of aluminum demand for machinery use and FRB index for 
nonelectrical machinery. 



16 



ALUMINUM DEMAND--DISSIPATIVE 

THOUSAND SHORT TONS < 1960-1976) 



Y 

600 



450 - 



300 - 



150 - 







I 




1 


"1 


/ 


- 




+ >4f 


^'^' 


- 


- 


• 






- 


, 


/m- 








^ • 




1 


1 









75 



150 



225 



300 X 



GROSS PRIVATE DOMESTIC INVESTMENT 
BILLION 1973 % 



Y = - 131.2343 + 
R-SQUARED = 0.7827 



2.6701 X 



PROJECTIONS 
1985 
2000 



707.1915 
1257.2416 



X 
314.0000 
520.0000 



FIGURE 7. - Scatter diagram of aluminum demand for dissipative use and gross private 
domestic investment. 



17 



ALUMINUM DEMAND— OTHER METAL 

THOUSAND SHORT TONS (1960-1976) 



Y 
4-00 



300 - 



200 - 



100 - 



L 





50 



100 



150 



FRBI -CHEMICALS AND PRODUCTS 
1967 = 100 



200 X 



Y = 176.3138 + ~ 
R-SQUARED = 0.1932 



0.6843 X 



PROJECTIONS Y 
1985 - 23.84.47 
2000 - 263.0086 



X 
292 . 5000 
642 . 0000 



FIGURE 8. - Scatter diagram of aluminum demand for otfier metal use and FRB index 
for chemicals and products. 



18 



ALUMINUM DEMAND— REFRACTORIES 

THOUSAND SHORT TONS < 1980-1976) 



Y 
300 



225 - 



150 - 



75 - 












50 



100 



150 



200 X 



FRBI-STONE, CLAY, AND GLASS 
1967 = 100 



Y = - 165.1793 + 
R-SQUARED = 0.7288 



2.9170 X 



PROJECTIONS 
1985 
2000 



342.0991 
616.5954. 



X 
173.9000 
268.0000 



FIGURE 9. - Scatter diagram of aluminum demand for refractories use and FRB index 
for stone, clay, and glass. 



19 



ALUMINUM DEMANDS-CHEMICALS 

THOUSAND SHORT TONS < 1960-1976) 



^l^w 


1 -■- -'T 


?— 








300 


r 




4- / 






200 


~ 


/ 


h 




•■ 


100 


j 


/ 






- 


r\ 


,/ 


L 




1 









50 



100 



150 



FRQI -PAPER AND PRODUCTS 
1967 = 100 



200 X 



Y = - 336.7368 + 
R- SQUARED = 0.8248 



5.2409 X 



PROJECTIONS 
1985 
2000 



Y 
725.5947 
1240.7756 



X 
202.7000 
301 .0000 



FIGURE 10. - Scatter diagram of aluminum demand for chemicals use and FRB index 
for paper and products. 



20 



ALUMINUM DEMAND — ABRASIVES 

THOUSAND SHORT TONS < 1960-1976) 



Y 
200 



150 - 



100 



50 - 












50 



100 



150 



200 X 



FRBI -TOTAL PRODUCTION 
1967 = 100 



Y = - 86.8939 + 
R-SQUARED = 0.6393 



1.5614 X 



PROJECTIONS 
1985 
2000 



Y 
203.9994. 
397.14-76 



X 

186.3000 
310.0000 



FIGURE 11. - Scatter diagram of aluminum demand for abrasives use and FRB index 
for total production. 



21 



The following tabulation summarizes the statistical projections and 
associated R^ values and explanatory variables: 



End Use 



Statistical 

projections, 

year 2000 

(thousand 

short tons) 



R^ value 



Explanatory variable 



Metal : 

Construction. . , 
Transportation , 
Electrical 



Cans and containers 

Appliances and equipment 



Machinery . . , 
Dissipative, 
Other , 



Nonmetal : 

Refractories, 



Chemicals . 
Abrasives, 



5,375 
2,033 
1,560 

1,942 
1,593 

980 

1,257 



617 

1,241 
397 



0.9549 
.9272 
.9562 

.9473 
.9647 

.8986 

.7827 

.1932 



,7288 

,8248 
,6393 



New construction activity. 
Motor vehicles index. -^ 
Major electrical equipment 

and parts index. -"^ 
Cans and containers index. ^ 
Gross private domestic 

investment . 
Nonelectrical machinery 

index . ^ 
Gross private domestic 

investment . 
Chemicals and products 

index . ^ 

Stone, clay, and glass 

index . ■'■ 
Paper and products index. ■'• 
Total production index. ^ 



^All indexes are Federal Reserve Board 
(1967 = 100). 



industrial production indexes 



In order to compare the historical consumption of each end use with the 
statistical projections for 1985 and year 2000, the statistical demand pro- 
jection system also provides the following computer graphs, figures 12 
through 22 . 



22 



ALUMINUM DEMAND— CONSTRUCTION 

THOUSAND SHORT TONS < 1960-1976) 

6000 



3000 - 



1 


1 


1 


+ 




— 1 

+ 


- 


_ -J... „ . 


. .L 


1 













1950 1960 1970 1980 1990 2000 2010 

FIGURE 12. - Aluminum demand for construction use. 



ALUMINUM DEMAND— TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 

3000 



1500 - 




1950 1960 1970 1980 1990 2000 2010 

FIGURE 13. - Aluminum demand for transportation use. 

ALUMINUM DEMAND--ELECTRICAL 

THOUSAND SHORT TONS (1960-1978) 





1 


1 


... . ._, — 




— r - ■ 


1 

+ 




■ 




_A 


/ 


+ 










...L 


•- 


« 






.L, 





2000 



1000 - 





■ "7 ■ -- 


1 


T 




' 


- T — 
















+ 




- 




_^ 


^ 


+ 






- 








1 






L 






1950 1960 1970 1980 1990 2000 2010 

FIGURE 14. - Aluminum demand for electrical use. 



ALUMINUM DEMAND — CANS AND CONTAINERS 
THOUSAND SHORT TONS <1960-1978> 



23 



Z000 



1000 - 




1950 1960 1970 1980 1990 2000 2010 

FIGURE 15. - Aluminum demand for cans and containers use. 

ALUMINUM DEMAND — APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS < 1960-1976) 



—r- — ■■ 


_.. _, 


r- ■ 


T 

4- 


4- 




- 




/ 






- 


L 


1, 


1 


• 


1 





2000 



1000 - 





.. , 


1- 








4- 




- 




-^^ 


/ 


4- 






- 




.. . L 


1 


1 




L 


— L 







1950 1960 1970 1980 1990 2000 2010 

FIGURE 16. - Alumjnum demand for appliances and equipment use. 



ALUMINUM DEMAND — MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



1000 



500 - 




1960 1970 1980 1990 2000 2010 

FIGURE 17. - Aluminum demand for machinery use. 



24 



ALUMINUM DEMAND — DISSIPATIVE 

THOUSAND SHORT TONS < 1960-1976) 

E000 





1 








— 1 


+ 












+ 










I . 


1. 


1 




, 


!_,,,. 





1000 - 





1950 1960 1970 1980 1990 2000 2010 

FIGURE 18. - Aluminum demand for dissipative use. 

ALUMINUM DEMAND — OTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



300 



150 - 




1950 




1960 1970 1980 1990 2000 

FIGURE 19. - Aluminum demand for other metal use. 



2010 



ALUMINUM DEMAND--REFRACTORIES 

THOUSAND SHORT TONS < 1960-1978) 



1000 



500 - 




1950 



T -■ 


1 




— 1 




■T ■ ■ 


1 ■ 


- 


-^^^^' 






-F 




4- 


, 


L. 




L 




1 


1 ■ 



1960 1970 1980 1990 2000 2010 

FIGURE 20. - Aluminum demand for refractories use. 



ALUMINUM DEriAND--CHEMICALS 

THOUSAND SHORT TONS (1960-19:^6) 



25 



1500 



750 





1 


J 


1 




1 


! 
4- 




- 








•4- 






- 




i 


L 


t 




.U 


1 






1950 i960 1970 1980 1990 2000 2010 

FIGURE 21. - Aluminum demand for chemicals use. 



ALUMINUM DEMANDS-ABRASIVES 

THOUSAND SHORT TONS (1960-1976) 



500 



250 - 











1 






1 


r - 

+ 




- 




/\ 




+ 






- 




L 








L 







1950 1960 1970 1980 1990 2000 2010 

FIGURE 22. - Aluminum demand for abrasives use. 



26 



CONCLUSION 

The statistical U.S. mineral end-use demand projection system described 
In this Information Circular Is only one component of the Bureau of Mines 
forecasting system. The results of statistical analyses obtained from the 
system provide very useful Information as to which related macroeconomlc 
variable best explains each historical pattern of end-use consumption, and, 
therefore, which macroeconomlc variable should be used as an explanatory 
factor for deriving each end-use statistical projection. To derive the pro- 
jection, a set of consistent estimated values for explanatory macroeconomlc 
variables is obtained from the Data Resources, Inc., macroeconomlc model. 
In the context of using regression analysis for projection purposes, it is 
assumed that the past empirical relationship will continue, but since there 
are many reasons this relationship might not hold true in the future, con- 
tingency analyses to derive a set of forecasts are necessary. Thus, the 
contingency analysis is also an important component of the Bureau of Mines 
demand forecasting system. 

Under the current system, the end-use statistical projections and con- 
tingency forecasts for each nonfuel mineral commodity in the year 2000 are 
updated and revised as new data or improved techniques become available. 



27 



APPENDIX A.~MACROECONOMIC VARIABLES USED IN BUREAU OF MINES 
STATISTICAL DEMAND PROJECTION SYSTEM 

1. U.S. gross national product 

2. U.S. population 

3. U.S. gross private domestic investment 

4. U.S. new construction activity 

5. The following Federal Reserve Board indexes of industrial production: 

Total production 
Textile mill products 
Paper and products 
Chemicals and products 

Basic chemicals 

Synthetic materials 

Paints 
Petroleum products 
Rubber and plastics products 

Tires 

Rubber excluding tires 

Plastics products, n.e.c. 
Stone, clay, and glass 
Iron and steel 

Basic steel and mill products 
Fabricated metal products 

Metal cans 

Hardware, plumbing, and structural metal 
Nonelectrical machinery 

Construction and allied equipment 

Metalworking machinery 

Special and general industrial equipment 
Electrical machinery 

Major electrical equipment and parts 

Household appliances 

Communication equipment 
Transportation equipment 

Motor vehicles and parts 
Automobiles 
Trucks, buses, and trailers 

Aircraft and parts 

Ships and boats 
Food and products 
Ordnance 



28 



APPENDIX B.— LEAST-SQUARES METHOD, BEST UNBIASED ESTIMATOR, 
AND MAXIMUM LIKELIHOOD ESTIMATOR 

The purpose of appendix B is to show how the least-squares formulas are 
obtained and how they are related to the "best unbiased estimator" and the 
"maximum likelihood" estimator. 

The simplest form of regression analysis assumes that the data points 
can be represented by the following linear equation: 

Y^ = a + b Xj + e^ (for t=l,2, ,n), (B-1) 

where Y^ = the dependent variable at time t, 

X^ = the explanatory (or independent) variable at time t, 

and e = the residual (or error term) at time t. 

In the formulation of equation B-1, two parameters, a and b, have to be esti- 
mated from the actual observations. These parameters can be estimated by many 
methods dependent upon the underlying assumptions and criteria. Three separate 
criteria often used are "least-squares," "best linear unbiased," and "maximum 
likelihood." 

The "Least-Squares" Criterion 

By the "least-squares" criterion, equation B-1 is fitted to the data 
points such that the sum of the squares of the residuals is minimized. From 
equation B-1, we have the sum of the squares of the residuals as follows: 

n n 

E e^ = S [Y - a-bX ]^. 
t=l t=l 

Therefore, the minimization of the sum of the squared residuals is equivalent 
to the minimization of the following term: 

n 

E [Y - a-bX ]^. (B-2) 

t=l * * 

Using the theory of calculus, the procedures for finding the estimates of 
a and b such that the value of equation B-2 is minimized are 

1. Taking the partial derivative of equation B-2 with respect to the 
parameters a and b respectively and setting them equal to zero: 

9(B-2) ^ Q (B_3) 

9,a 
and 

iif=2i - 0. (B-.) 



29 



2. Obtaining the estimate of the parameters a and b by solving equations 
B— 3 and B-4 simultaneously. 

By procedure 1, we have 

n 
-2 E [Y^ - a-bXj ] = (B-5) 

t=l 

and 

n 
-2 E [Y - a-bX ] X =0. (B-6) 

t=l * * * 



Solving for a from equation B-5, we get 



n 
a = (1/n) S [Y - bX ] . (B-7) 

t=l * * 

Substituting equation B-7 into B-6 and solving for b, we obtain the estimate 
of b as follows: 

y^ n n n n n n 

b = [ E X^Y^ - (1/n) E X^ E Yj/[ E X2 - (1/n) E X^ ^ \] 
t=l t=l t=l t=l t=l t=l 

n _ _ n 

=[E XY -nXY]/[E X^-n X^] (B-8) 

t=l * ' t=l ^ 

_ " _ n 
where X = (1/n) E X , and Y = (1/n) E Y . (B-9) 

t=l ' t=l * 

Equation B-7 can be written 

a = Y - bX. (B-10) 

After the estimate of b is obtained from formula B-8, the estimate of the 
parameter a can be obtained by substituting b for b in equation B-10: 

A - A- , ^ 

a = Y - bX. (B-11) 

The "Best Linear Unbiased" Criterion 

To apply the "best linear unbiased" criterion, it is necessary to make 
the following assumptions about the residuals in equation B-1 because "best 
linear unbiased" is a statistical concept: 



30 



E[e^ ] = 0, for t=l,2, 3, . . . ,n (B-12) 

for t^j ; t,j=l,2,3, .. .,n 
and E[e^ e. ] = < (B-13) 

' S^ for t=j; t,j=l,2,3, ...,n. 

Condition B-12 assumes the expected value of the residuals is zero for each t, 
and condition B-13 assumes that all the residuals have the same variance S^ 
and zero covariance between different residuals. By the "linear" criterion, 
the estimate of b is a linear function of the observations on Y, and it can be 
expressed as follows: 

n 
b = E w^Y^ . (B-14) 

t=l 

Equation B-14 is a class of linear functions on Y with weights w^ . According 
to the "best linear unbiased" criterion, these weights are chosen to satisfy 
the following conditions: 

1. The "unbiased" condition — The expected value of the estimator should 
be equal to the true parameter to be estimated; that is, 

A 
E[b] = b. (B-15) 

2 . The "best" condition — The variance of the estimator should be 
minimized; that is. 

Minimize V(b). (B-16) 

Combining equations B-1 and B-14, the linear estimator of b can be rewritten 

A n 
b = E w, Y, 
t=l * 

n 

= Z w (a + bX + e ) 
t=l ' 

n n n 
= a Z w^ + b E w^ X^ + Z w^ e^ • (B-17) 

t=l t=l t^l 

By taking the expected value of equation B-17, we have 

A n n n 

E[b] = E[a E w +b E w^ X^ + E w^ e^ ] 
t-1 t=l t=l 

n n n 

= a E w^^-b E \]^ X^ + E Wt E[et ] . (B-18) 

t=l t=l t=l 



31 



Assumption B-12 makes the last term of B-18 equal to zero. Thus, the expected 
value of the estimator is 

An n 
E[b] =a Z w^+b Z w^Xj. (B-19) 

t=l t=l 

By comparing equation B-19 with the "unbiased" condition (B-15) , it is shown 

A 
that b is a linear unbiased estimator of b if the following conditions are 
satisfied: 

n 

Z w^ = (B-20) 

t=l 

n 
and Z w^X^ = 1. (B-21) 

t=l 

Using assumption B-13 and equation B-17, the variance of b is 

A n 
V(b) = V( Z we.) 
t=l 

n 
= S2 z w2. (B-22) 

t=l 

A 
Since S is a given constant, the value of V(b) will be minimized if the 
weights are chosen in such a way that the sum of the squares of the weights 
is minimized. Thus, the construction of a "best linear unbiased" estimator 
of the parameter b is simply to find a set of weights such that conditions 

n 
B-20 and B-21 are satisfied and Z w is minimized. 

t=l ' 



This constrained minimization problem can be solved by minimizing the follow- 
ing function: 

n n n 

F = Z w^^ - 2K Z w^ - 2H( Z w^ Xj - 1) , (B-23) 

t=l t=l t=l 



where K and H are Lagrangian multipliers. 



32 



Differentiating F with respect to w^ , K, and H and setting them equal to zero 
gives the equations 



w^ = K + HX^ for t=l,2,3, ...,n, (B-24) 



n 



nK + H E X . (B-27) 



2 w = 0, (B-25) 

t=l 

n 
and E w^X^ =1. (B-26) 

t=l 

Summing equation B-24 over t yields 

n n 

Z w^ = Z (K + HX^ ) 
t=l t=l 

n 

E 

t=l 

Substituting condition B-25 into the left side of B-27, we have 

n 
nK + H Z X =0 
t=l ' 

n 
nK = - H S X 
t=l * 

n 
K = - H (1/n) E X 
t=l * 

K = - H X. (B-28) 

Combining equations B-28 and B-24 yields 

w =-HX + HX 
t t 

= H(X^ -X) . (B-29) 

By multiplying both sides of equation B-29 by X and siomming over all t, 
we get 

n n _ 

E w X = E [H(X - X)X ]. (B-30) 

t=l t * t=l * * 



33 



By condition B-26, the left side of equation B-30 is equal to 1: 

n 
1 = E [H(Xt - X)\ ] 
t=l 

n 

1 = H E [X,^ - X X 1 
t t 

t=l 

n _ n 

1 = H[E X^-X E X] 
t=l ' t=l * 

n _ 

H = l/[ E X2 - n X^]. (B-31) 

t=l * 



From equations B-14 and B-29, we have 



n 



b = E w^Y^ 
t=l 

n _ 

= E H (X - X)Y 
t=l t * 

n _ 

= H E (X Y - XY ) 
t=l * * 

n _ _ 

=H[E XY -nXY] (B-32) 

t=l * ' 

Finally, the "best linear unbiased" estimator of the parameter b can be 
obtained by substituting equation B-31 into B-32 : 

An n 

b=[E XY -nX Y]/[ E X^ - n X^] . (B-33) 

t=l * * t=l * 

Comparing equations B-33 and B-8 shows that the formula for calculating the 
"best linear unbiased" estimate of the parameter b is exactly the same as the 
formula used for the "least-squares" estimation. 

The "Maximum Likelihood" Criterion 

Under the "best linear unbiased" criterion, the probability distribution 
of the residuals is not specified. If the probability distribution of the 



34 



residuals is specified explicitly to be a "normal distribution," the combina- 
tion of this "normality" assumption with conditions B-1, B-12, and B-13 yields 
the following likelihood function L: 

( - Vzrt ) =- 1 '^ 

L(a,b,s2) = (2^S^) exp [-(2S2) E (Y - a-bX )2] (B-34) 

t=l * * 

where it = 3.14159 , 

and exp = exponential. 

The above likelihood function gives the probability density at the sample data 
points (Y^ ,X^ ) . For a given set of the data points, the function L becomes a 
function of the parameters a, b, and S^. Consequently, the problem of finding 
a "maximum likelihood" estimator of the parameters a, b, and S^ is the problem 
of finding an estimate of the parameters that maximizes the probability of 
obtaining the given observed data points. 

The function L (a,b,S^) can be maximized by setting the partial deriva^ 
tive of In L, with respect to the parameters a, b, and S^, equal to zero and 
solving the resulting equations for the parameters a, b, and S^: 

o 1 n 
In L = - J^n In 277 - Jgn In S2 - (2s2) E (Y^ - a-bX^ )^ (B-35) 

t=l 

where In = natural logarithm. 



t t 



^i^^ = (S^)"^ E (Y - a-bX ) = 0. (B-36) 



3a ^ ' t=l * 



SI n T - 1 ^ 

—.T — = (S^) E (Y - a-bX )X = 0, 

oD t t t 



(B-37) 



1 - 1 



n 



_81nL = -n(2s2)" + (2S'+)' E (Y, - a-bK )2 = 0. (B-38) 

8S2 t=l 

Comparing equations B-5 and B-6 with equations B-36 and B-37 indicates that 
the "maximum likelihood" estimator is a "least-squares" estimator. Similarly, 
it can be easily shown that the "maximum likelihood" estimator is also a "best 
linear unbiased" estimator. In summary, the following important properties 
and implications are established from the preceding derivations: 

1. Under the strict "least-squares" criterion, it is not necessary to 
assume that the residuals are statistically "random variables." Without this 
statistical assumption, the "least-squares" method is simply a mathematical 
method of fitting a linear equation to a set of observed data points (Y^ ,X^ ) 
so that the sum of the squared distances from the data points to the 
fitted equation will be minimized. 



35 



2. Under the "best linear unbiased" criterion, statistical assumptions 
regarding the residuals are required. Given the statistical assumptions B-12 
and B-13, the "best linear unbiased" estimator is also a "least-squares" 
estimator. On the other hand, excepting the case in which the conditions 
B-12 and B-13 are imposed upon the residuals, the "least-squares" estimator 
is not always a "best linear unbiased" estimator. If assumptions B-12 and 
B-13 are satisfied, the method of "least-squares" provides the best available 
estimate in the sense that the variance of the estimator is smaller than, or 
at most equal to, the variances of all other "linear unbiased" estimators, 
and this holds for any probability distribution of the residuals. 

3. Under conditions B-12 and B-13 and with the additional assumption of 
the "normality" of the residuals, the "maximum likelihood" estimator is a 
"best linear unbiased" estimator and it is also a "least-squares" estimator. 
Without the "normality" assumption for the residuals, the "maximum likelihood" 
estimator is not necessarily a "best linear unbiased" estimator. Hence, if a 
"best linear unbiased" estimator is desired, it will require the "normality" 
assumption under the "maximum likelihood" criterion. 



36 



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37 



APPENDIX D.— DATA FOR MACROECONOMIC VARIABLES 
USED TO DERIVE STATISTICAL PROJECTIONS 





U.S. 




U.S. 


U.S. new 




gross national 


U.S. 


gross private 


construction 


Year 


product, 


population, 


domestic 


activity. 




billion 


million persons 


investment. 


billion 




1972 dollars 




billion 
1973 dollars 


1973 dollars 


1960 


736.8 


181 


117.70 


84.27 


1961 


755.3 


184 


113.47 


86.13 


1962 


799.1 


187 


127.77 


90.28 


1963 


830.7 


189 


133.29 


95.76 


1964 


874.4 


192 


140.57 


98.52 


1965 


925.9 


194 


159.43 


104.91 


1966 


981.0 


197 


171.61 


105.31 


1967 


1,007.7 


199 


161.74 


104.57 


1968 


1,051.8 


201 


168.50 


111.61 


1969 


1,078.8 


203 


178.36 


114.55 


1970 


1,075.3 


205 


163.06 


109.90 


1971 


1,107.5 


207 


176.29 


121.20 


1972 


1,171.1 


209 


199.22 


131.29 


1973 


1,235.0 


210 


220.00 


137.90 


1974 


1,214.0 


212 


195.40 


125.87 


1975 


1,191.1 


214 


152.74 


111.67 


1976 


1,264.7 


215.1 


190.73 


116.67 





Federal Re 


serve Board industrial production indexes (1967=100) 


Year 


Total 


Textile mill 


Paper and 


Chemicals and 


Basic 




production 


products 


products 


products 


chemicals 


1960 


66.2 


69.3 


68.0 


56.4 


60.9 


1961 


66.7 


71.4 


71.8 


59.2 


63.2 


1962 


72.2 


76.2 


76.2 


65.7 


70.1 


1963 


76.5 


78.9 


80.6 


71.8 


77.3 


1964 


81.7 


85.2 


85.4 


78.8 


85.2 


1965 


89.8 


92.8 


91.9 


87.8 


93.8 


1966 


97.8 


98.4 


99.2 


95.7 


99.6 


1967 


100.0 


100.0 


100.0 


100.0 


100.0 


1968 


106.3 


107.9 


107.3 


109.5 


107.4 


1969 


111.1 


112.6 


115.8 


118.4 


117.0 


1970 


107.8 


111.8 


115.2 


120.4 


117.9 


1971 


109.6 


116.5 


120.4 


125.9 


123.7 


1972 


119.7 


132.7 


128.8 


143.6 


137.7 


1973 


129.8 


142.9 


137.4 


154.5 


147.7 


1974 


129.3 


132.8 


134.5 


159.4 


153.3 


1975 


117.8 


122.3 


116.3 


147.2 


135.9 


1976 


129.8 


135.9 


133.1 


169.4 


158.5 



38 





Federal 


Reserve Board industrial 


production indexes (1967 


=100) 


Year 


Synthetic 


Paints 


Petroleum 


Rubber and plastics 


Tires 




materials 




products 


products 




1960 


42.9 


72.9 


76.7 


52.2 


76.6 


1961 


46.3 


71.0 


79.8 


54.5 


73.5 


1962 


54.8 


75.1 


84.0 


61.8 


83.5 


1963 


60.8 


82.1 


87.9 


66.4 


86.4 


1964 


70.8 


87.2 


90.3 


75.3 


95.7 


1965 


85.2 


95.3 


92.1 


86.2 


101.8 


1966 


96.7 


100.3 


96.1 


96.8 


109.7 


1967 


100.0 


100.0 


100.0 


100.0 


100.0 


1968 


126.9 


107.0 


105.1 


119.7 


122.1 


1969 


141.2 


112.0 


108.4 


130.2 


124.0 


1970 


145.1 


105.6 


113.2 


132.3 


112.2 


1971 


160.4 


111.1 


116.7 


147.6 


125.7 


1972 


194.7 


118.0 


122.1 


172.4 


141.9 


1973 


221.1 


114.7 


128.5 


184.1 


141.2 


1974 


224.4 


119.2 


124.7 


195.2 


147.9 


1975 


191.0 


114.2 


124.1 


166.7 


124.7 


1976 


232.6 


124.2 


132.7 


200.2 


127.9 





Federal Reserve Board industrial production indexes ( 


:i967=100) 


Year 


Rubber 


Plastics 


Stone, clay. 


Iron and 


Basic steel 




excluding 


products, 


and glass 


steel 


and mill 




tires 


n.e.c. 






products 


1960 


66.0 


32.0 


79.0 


77.7 


79.3 


1961 


68.7 


36.6 


78.7 


74.2 


76.1 


1962 


77.4 


41.5 


83.5 


77.3 


78.2 


1963 


80.8 


47.6 


88.5 


84.3 


84.6 


1964 


88.1 


57.0 


92.7 


95.9 


96.3 


1965 


94.7 


72.2 


98.3 


105.2 


105.2 


1966 


101.6 


85.9 


101.1 


108.4 


107.8 


1967 


100.0 


100.0 


100.0 


100.0 


100.0 


1968 


107.2 


126.7 


106.3 


103.2 


103.3 


1969 


115.2 


144.1 


109.6 


112.6 


112.9 


1970 


107.7 


161.4 


106.0 


104.7 


106.5 


1971 


114.1 


183.8 


110.9 


96.1 


95.9 


1972 


127.6 


221.5 


120.8 


107.1 


107.0 


1973 


144.1 


237.5 


133.5 


122.3 


122.9 


1974 


141.3 


260.7 


133.1 


119.8 


121.6 


1975 


121.5 ' 


223.2 


117.8 


95.8 


96.8 


1976 


143.8 


281.9 


135.8 


104.4 


105.4 



39 





Federal Reserve Board industrial production indexes (1967=100) 


Year 


Fabricated 


Metal 


Hardware, plumbing. 


Nonelectrical 


Construction and 




metal 


cans 


and structural metal 


machinery 


allied equipment 




products 










1960 


71.1 


74.8 


69.8 


56.9 


65.4 


1961 


69.4 


79.5 


68.3 


55.4 


59.0 


1962 


75.4 


78.5 


73.0 


62.1 


65.4 


1963 


77.8 


76.8 


76.2 


66.3 


72.3 


1964 


82.6 


82.3 


81.6 


75.6 


86.6 


1965 


90.8 


86.0 


90.0 


85.0 


97.0 


1966 


97.2 


90.4 


96.6 


98.8 


104.7 


1967 


100.0 


100.0 


100.0 


100.0 


100.0 


1968 


105.6 


110.3 


103.4 


101.8 


104.5 


1969 


107.9 


113.2 


106.8 


109.3 


107.4 


1970 


102.4 


120.9 


105.2 


104.4 


99.0 


1971 


103.5 


114.4 


107.9 


100.2 


100.6 


1972 


112.1 


119.0 


118.3 


116.0 


112.6 


1973 


124.7 


126.5 


131.1 


133.7 


127.8 


1974 


124.2 


132.9 


132.8 


140.1 


137.8 


1975 


109.9 


126.5 


120.3 


125.1 


129.3 


1976 


123.3 


136.7 


130.0 


134.7 


137.9 





Federal Res 


erve Board industrial production indexes (1967=100) 






Special 




Major 




Year 


Metalworking 


and general 


Electrical 


electrical 


Household 




machinery 


industrial 
equipment 


machinery 


equipment 
and parts 


appliances 


1960 


60.5 


62.2 


51.6 


62.1 


59.4 


1961 


56.3 


61.5 


54.6 


61.9 


61.8 


1962 


66.7 


67.3 


62.9 


67.1 


68.1 


1963 


67.6 


71.5 


64.7 


68.8 


79.9 


1964 


76.5 


80.9 


68.4 


75.5 


82.8 


1965 


86.3 


91.6 


81.7 


86.2 


92.7 


1966 


98.9 


102.7 


97.9 


99.5 


99.2 


1967 


100.0 


100.0 


100.0 


100.0 


100.0 


1968 


94.2 


98.5 


105.5 


101.0 


108.7 


1969 


93.9 


102.8 


111.9 


106.8 


115.2 


1970 


85.6 


95.6 


108.1 


105.0 


117.4 


1971 


74.1 


88.4 


107.7 


98.6 


124.8 


1972 


85.7 


96.9 


122.2 


113.5 


145.4 


1973 


101.0 


115.2 


143.1 


131.3 


155.8 


1974 


104.2 


118.3 


143.8 


135.7 


146.3 


1975 


93.0 


104.1 


116.5 


106.1 


117.8 


1976 


98.3 


111.2 


131.7 


114.5 


128.4 



40 





Federal Reserve Board Indus 


trial production indexes 


(1967=100) 


Year 


Coiranunication 


Transportation 


Motor 




Trucks, buses. 




equipment 


equipment 


vehicles 
and parts 


Automobiles 


and trailers 


1960 


56.1 


65.4 


74.7 


75.4 


65.8 


1961 


66.3 


61.5 


65.5 


64.0 


64.4 


1962 


80.6 


71.1 


79.8 


82.1 


71.9 


1963 


77.6 


78.0 


88.3 


91.4 


85.8 


1964 


74.2 


80.0 


90.7 


92.9 


90.7 


1965 


81.6 


95.1 


115.9 


121.0 


111.3 


1966 


92.5 


102.0 


113.9 


116.1 


114.4 


1967 


100.0 


100.0 


100.0 


100.0 


100.0 


1968 


109.5 


111.1 


120.3 


120.9 


122.6 


1969 


111.5 


108.4 


116.5 


113.8 


123.7 


1970 


105.8 


89.5 


92.3 


86.6 


102.0 


1971 


97.5 


97.9 


118.6 


116.0 


123.4 


1972 


101.4 


108.2 


135.8 


128.6 


154.2 


1973 


108.8 


118.3 


148.8 


138.3 


181.5 


1974 


114.1 


108.7 


128.2 


107.9 


172.4 


1975 


105.6 


97.4 


111.1 


101.1 


140.6 


1976 


107.7 


110.6 


140.7 


132.0 


NA^ 



•'•The trucks, buses, and t 
because the classifica 



railers index 
tion has been 



number is 
changed . 



not available after 1975 



Year 


Federal Reserve 


Board industrial production indexes (1967=100) 




Aircraft and parts 


Ships and boats 


Food and products 


Ordnance 


1960 


55.9 


63.7 


78.6 


50.1 


1961 


58.3 


65.8 


80.9 


42.3 


1962 


64.0 


68.2 


83.4 


48.6 


1963 


68.8 


71.3 


86.4 


55.7 


1964 


67.7 


77.3 


90.4 


54.9 


1965 


71.9 


86.0 


92.4 


66.7 


1966 


88.4 


94.1 


96.0 


82.9 


1967 


100.0 


99.9 


100.0 


100.0 


1968 


103.0 


97.6 


102.6 


115.6 


1969 


97.3 


98.7 


106.1 


111.9 


1970 


81.1 


94.7 


108.9 


92.7 


1971 


67.7 


95.3 


112.8 


85.1 


1972 


67.0 


114.1 


116.8 


84.2 


1973 


75.1 


120.7 


120.9 


81.3 


1974 


78.3 


129.4 


124.0 


78.9 


1975 


73.3 


133.6 


123.4 


76.6 


1976 


69.3 


151.3 


132.3 


72.7 



41 



APPENDIX E.— MACROECONOMIC EXPLANATORY VARIABLES USED IN ESTIMATED ALUMINUM 
END-USE CONSUMPTION EQUATIONS AND ASSOCIATED R^ VALUES 

The number shown after each macroeconomic variable is the R^ value: 
* indicates the best R value. 

Construction 

GNP 0.8280 

U.S. population 0.7501 
*New construction activity 0.9549 
FRB index — Total production 0.8519 

FRB index — Construction and allied equipment 0.7205 
FRB index — Hardware, plumbing, and structural metal 0.8557 

Transportation 

GNP 0.7776 

FRB index — Total production 0.8197 
FRB index— Tires 0.8078 
FRB index — Automobiles 0.8172 
FRB index — Transportation equipment 0.8796 
FRB index — Petroleum products 0.7135 
*FRB index— Vehicles 0.9272 
FRB index — Trucks, buses, and trailers 0.8941 
FRB index — Aircraft and parts 0.1554 
FRB index — Ships and boats 0.6762 

Electrical 
GNP 0.8260 

FRB index — Total production 0.8771 
FRB index — Communication equipment 0.7993 
*FRB index — ^Major electrical equipment and parts 0.9562 
FRB index — Electrical machinery 0.9235 
FRB index — Household appliances 0.9257 
FRB index — Fabricated metal products 0.8899 

Cans and containers 



GNP 0.9271 

FRB index — Total production 0.9089 
FRB index — Metalworking machinery 0.4706 
FRB index — Fabricated metal products 0.8778. 
*FRB index — ^Metal cans 0.9473 

Appliances and equipment 

GNP 0.7404 

U.S. population 0.6436 
*Gross private domestic investment (GPDI) 0.9647 
FRB index — Total production 0.7974 

FRB index — Major electrical equipment and parts 0.8491 
FRB index — Electrical machinery 0.8300 
FRB index — Household appliances 0.8757 
FRB index — Communication equipment 0.7009 



42 



Machinery 

GNP 0.8373 

U.S. population 0.7606 
FRB index — Total production 0.8844 
*FRB index — Nonelectrical machinery 0.8986 
FRB index — Metalworking machinery 0.7098 
FRB index — Special and general industrial machinery 0.8601 

Dissipative 

GNP 0.6249 

U.S. population 0.5443 
*Gross private domestic investment (GPDI) 0.7827 
FRB index — Total production 0.6836 
FRB index — Chemicals and products 0.5267 
FRB index — Basic chemicals 0.5572 

Other Metal 

GNP 0.1135 

U.S. population 0.1271 
GPDI 0.0231 

FRB index — Total production 0.0893 
FRB index— Paints 0.0516 
*FRB index — Chemicals and products 0.1932 
FRB index — Basic chemicals 0.1731 

Refractories 

GNP 0.6014 

U.S. population 0.4693 
GPDI 0.4241 

FRB index — Total production 0.6840 
*FRB index — Stone, clay, and glass 0.7288 
FRB index — Iron and steel 0.3764 
FRB index — Basic steel and mill products 0.4083 

Chemicals 

GNP 0.7079 

U.S. population 0.7181 
GPDI 0.3319 

FRB index — Total production 0.6727 
FRB index — Basic chemicals 0.7406 
FRB index — Chemicals and products 0.7185 
FRB index — Plastics 0.7816 
*FRB index— Paper 0.8248 
FRB index — Textiles 0.7448 

Abrasives 

GNP 0.4932 

U.S. population 0.3406 
GPDI 0.5201 
*FRB index — Total production 0.6393 
FRB index — Iron and steel 0.5707 
FRB index — Metalworking machinery 0.3077 
FRB index — Basic steel 0.6114 



43 



APPENDIX F. —OTHER COMPUTER GRAPHS WITH LOWER R^ VALUES 



ALUMINUM DEMAND-CONSTRUCTIOrj 

THOUSAND SHORT TONS < 1960-1976) 



Y 



^kJiOkJ 


! 


1 

+ 


1 y 




1500 










1000 


"■ 


At + 




■■ 


500 


J 






- 


n 


/l 


I 


1 





500 1000 
GNP (BILLION 1972 $> 



1500 



2000 X 



Y = - 764.6077 + 
R- SQUARED = 0.8280 



1.7182 X 



PROJECTIONS 
1985 
2000 



Y 
2277.5807 
4032.7886 



X 
17 70.5000 
2792.0000 



FIGURE F-1. - Scatter diagram of aluminum demand for construction use and gross 
national product (GNP). 



44 



ALUMINUM DEMAND~-CONSTRUCTION 

THOUSAND SHORT TONS < 1960-1976) 



Y 
2000 



1500 - 



1000 



500 - 



i- 





75 150 225 300 X 
U.S. POPULATION (MILLION PERSONS) 



Y = ^ 4270.04-55 + 
R-SQUARED = 0.7501 



26.2771 X 



PROJECTIONS 
1985 
2000 



1881. 4. 4-30 
2627. 7151 



X 

234.1000 
262 . 5000 



FIGURE F-2. - Scatter diagram of aluminum demand for construction use and U.S. 
population. 



45 



ALUMINUM DEMAWD--CONSTRUCTION 

THOUSAND SHORT TONS < 1960-1976) 




150 



200 X 



FRBI -TOTAL PRODUCTION 
1967 = 100 



Y - - 401.3510 + 
R-SQUARED = 0.8519 



13.7556 X 



PROJECTIONS Y X 

1985 2161.3177 186.3000 
2000 3862.8858 310.0000 

FIGURE F-3. - Scatter diagram of aluminum demand for construction use and FRB 
index for total production. 



46 



ALUMINUM DEMAND— CONSTRUCTION 

THOUSAND SHORT TONS < 1960-1976) 



Y 
2000 



1500 - 



1000 - 



300 - 












50 



100 



150 



200 X 



FRBI-CONSTRUCTION & ALLIED EQUIPMENT 
1967 = 100 



Y = - 14-0.734.9 + 
R-SQUARED = 0.7205 



11.1992 X 



PROJECTIONS 
1985 
2000 



Y 
1974-. 804-9 
2699.3969 



X 
188.9000 
253.6000 



FIGURE F-4. - Scatter diagram of aluminum demand for construction use and FRB 
index for construction and allied equipment. 



\"'m 



47 



ALUMINUM DEMANDS-CONSTRUCTION 

THOUSAND SHORT TONS < 1960-1976) 



Y 

2000 



1500 - 



1000 - 



500 - 












50 



100 



150 



200 X 



FRBI-HARDWARE, PLUMBING, STRUCTURAL METAL 
1987 = 100 



Y = - 4.19.8344 + 
R-SQUARED = 0.8557 



13.94-56 X 



PROJECTIONS Y X 

1985 2214.4-983 188.9000 
2000 3401.2728 274.0000 

FIGURE F-5. - Scatter diagram of aluminum demand for construction use and FRB 
index for hardware, plumbing, structural metal. 



48 



ALUMINUM DEMAND-TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 



L- 




500 1000 
GNP (BILLION 1972 $) 



1500 



2000 X 



Y = -. 323.7238 + 
R-SQUARED = 0.7776 



0.9533 X 



PROJECTIONS Y 
1985 1364.1987 
2000 2338.0552 



X 
17 70.5000 
2792.0000 



FIGURE F-6. - Scatter diagram of aluminum demand for transportation use and gross 
national product (GNP). 



49 



ALUMINUM DEMAND- -TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 



y 

1200 



900 - 



600 - 



300 



•- 





50 



100 



150 



200 X 



FRBI -TOTAL PRODUCTION 
1967 = 100 



Y = - 131.5764 + 
R-SQUARED = 0.8197 



7.7254- X 



PROJECTIONS Y X 

1985 1307.6810 186.3000 
2000 2263 . 3233 310. 0000 

FIGURE F-7. - Scatter diagram of aluminum demand for transportation use and FRB 
index for total production. 



50 



rLUMINUM DEMANDS-TRANSPORTATION 

THOUSAND SHORT TONS (1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 



^ 





50 



100 



150 



200 X 



FRBI-TIRES 
1967 = 100 



Y = - 160.6120 + 
R-SQUARED = 0.8078 



7.24-10 X 



PROJECTIONS 
1985 
2000 



1175.3678 
1708.3115 



X 
184.. 5000 
258.1000 



FIGIJRE F-8. - Scatter diagram of aluminum demand for transportation use and FRB 
index for tires. 



A_:i 



51 



ALUMINUM DEMANDS-TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 












50 



100 



150 



200 X 



FRBI -AUTOMOBILES 
1967 = 100 



Y = - 201.1124. + 
R-SQUARED = 0.8172 



8.0582 X 



PROJECTIONS 
1985 
2000 



1008.4270 
1402.4.74-3 



X 
150.1000 
199.0000 



FIGURE F-9. - Scatter diagram of aluminum demand for transportation use and FRB 
index for automobiles. 



52 



ALUMINUM DEMAND--TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 



Y 

1200 



900 - 



600 - 



300 - 












50 



100 



150 



200 X 



FRBI -TRANSPORTATION EQUIPMENT 
1967 = 100 



Y = - 309.24-35 + 
R-SQUARED = 0.8796 



10.134-1 X 



PROJECTIONS Y X 

1985 1227.1003 151.6000 
2000 2629.6728 290.0000 

FIGURE F-10. - Scatter diagram of aluminum demand for transportation use and FRB 
index for transportation equipment. 



53 



ALUMINUM DEMAND-TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 












50 



100 



150 



200 X 



FRBI -PETROLEUM PRODUCTS 
1967 = 100 



Y = - 267.9487 + 
R-SQUARED = 0.7135 



8.7214. X 



PROJECTIONS 
1985 
2000 

FIGURE F-n 



Y 
1306.274-6 
24.09.5391 



X 
180.5000 
307.0000 



Scatter diagram of aluminum demand for transportation use and FRB 
index for petroleum products. 



54 



ALUMINUM DEMAND--TRANSPORTATION 
THOUSAND SHORT TONS 



Y 

1200 



900 - 



B00 - 



300 - 












50 



100 



150 



200 X 



FRBI-TRUCKS, BUSES. & TRAILERS 
1967 = 100 



Y = 76.9576 + 
R-SQUARED = 0.894-1 



^.8619 X 



PROJECTIONS 
1985 
2000 



Y 
1309.4727 
217 7.3385 



253.5000 
432 . 0000 



FIGURE F-12. - Scatter diagram of aluminum demand for transportation use and FRB 
index for trucks, buses, and trailers. 



55 



ALUMINUM DEMAND~-TRANSPORTATION 

THOUSAND SHORT TONS < 1960-1976) 



^161 u:i 




1 

+ 


1 


/ 


/ 


900 




+ + 


^+ 






600 




+ 


+ 






300 


^ 


+ 








171 






1 


1 









50 



100 



150 



200 X 



FRBI-AIRCRAFT & PARTS 
1967 = 100 



Y = 24-8.74-05 + 
R-SQUARED = 0.1554- 



5.2532 X 



PROJECTIONS 
1985 
2000 



Y 
857 . 5879 
1315.6681 



X 
115.9000 
203.1000 



FIGURE F-13. - Scatter diagram of aluminum demand for transportation use and FRB 
index for aircraft and parts. 



56 



ALUMINUM DEMAND~-TRANSPORTATION 

THOUSAND SHORT TONS < 1960- 1976 > 



Y 
1200 



900 - 



600 - 



300 - 












50 



100 



150 



200 X 



FRBI-SHIPS & BOATS 
1967 = 100 



Y = 54.. 5074 + 
R~SQUARED ^ 0.6762 



6.0560 X 



PROJECTIONS 
1985 
2000 



878.1331 
921 . 7368 



X 
136.0000 
14-3.2000 



FIGURE F-14. - Scatter diagram of aluminum demand for transportation use and FRB 
index for ships and boats. 



57 



ALUMINUM DEMAND-ELECTRICAL 

THOUSAND SHORT TONS (1960-1976) 



Y 

1000 



750 - 



500 - 



250 - 








500 1000 
GNP (BILLION 1972 $) 



1500 



2000 X 



Y - - 4-29.3296 + 
R-SQUARED = 0.8260 



0.9443 X 



PROJECTIONS Y X 

1985 1242.5560 17 70.5000 
2000 2207.1599 2792.0000 

FIGURE F-15. - Scatter diagram of aluminum demand for electrical use and gross 
national product (GNP). 



58 



ALUMINUM DEMAND--ELECTRICAL 

THOUSAND SHORT TONS (1960-1976) 



Y 
1000 



750 - 



500 - 



250 - 












50 



100 



150 



200 X 



FRBI -TOTAL PRODUCTION 
1967 = 100 



Y = - 241.8019 + 
R-SQUARED = 0.8771 



7.6798 X 



PROJECTIONS 
1985 
2000 



Y 
1188.94-99 
2138.94-47 



186.3000 
310.0000 



FIGURE F-16. - Scatter diagram of aluminum demand for electrical use and FRB 
index for total production. 



59 



ALUMINUM DEMAND-~ELECTRICAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 
1000 



750 - 



500 - 



250 



»■ 








++ / 


/ 








+ /+ 

+ A 


















V 






- 






L. .. . 


1 





50 



100 



150 



200 X 



FRBI-COMMUNICATION EQUIPMENT 
1967 = 100 



Y = - 323.7736 4- 
R-SQUARED = 0.7993 



9.1414. X 



PROJECTIONS 
1985 
2000 



Y 
1062.0638 
1842.7401 



151.6000 
237.0000 



FIGURE F-17. - Scatter diagram of aluminum demand for electrical use and FRB 
index for communication equipment. 



60 



ALUMINUM DEMAND~-ELECTRICAL 

THOUSAND SHORT TONS (1960-1976) 



Y 
1000 



750 - 



500 - 



250 












50 



100 



150 



200 X 



FRBI-ELECTRICAL MACHINERY 
1967 = 100 



Y = - 38.7270 + 
R -SQUARED = 0.9235 



5.7982 X 



PROJECTIONS Y X 

1985 987.5575 177.0000 
2000 1410.8273 250.0000 

FIGURE F-18. - Scatter diagram of aluminum demand for electrical use and FRB 
index for electrical machinery. 



ri.'-'-,TSTlS;;{l.';SSfiitSi5 



,..U .j,IJW.< il«IIHg-ri...ai,„t<,rnH.IW"l4-llU!.'.' 



ai'M!iaaMiiaaaMtoMa!«afieMiEMitiiiUiiiia«?iiMMiiiiM « Qtii M i ! 8iii M HB^^ 



61 



ALUMINUM DEMAND~-ELECTRICAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 

1000 



750 - 



500 - 



250 - 



L 





50 



100 



150 



FRBI -HOUSEHOLD APPLIANCES 
1967 = 100 



200 X 



Y = - 89.7623 4- 
R-SQUARED = 0.9257 



5.8568 X 



PROJECTIONS Y X 

1985 1539.0213 278.1000 
2000 2779.4-973 4-89.9000 

FIGURE F-19. - Scatter diagram of aluminum demand for electrical use and FRB 
index for household appliances. 



IP- 



"msmmmm, 



W^^M^T^riSu^^^iS?&ISW^^^a!lS 



StiHEH^Safn-JMKE: 



,dii5aii^ia:ii::i|;.'Bi)fife;i3ii!j«ii!iffi*iiij!ifli.:i»ri!Wji(ii^^^^^ 



62 



ALUMINUM DEMAND—ELECTRICAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 
1000 



750 - 



500 - 



250 - 












50 



100 



150 



200 X 



FRBI-FABRICATED METAL PRODUCTS 
1967 =100 



Y = - 384.0102 + 
R-SQUARED = 0.8899 



9.2771 X 



PROJECTIONS 
1985 
2000 



1255.2668 
2028 . 0556 



X 
176.7000 
260 . 0000 



FIGURE F-20. - Scatter diagram of aluminum demand for electrical use and FRB 
index for fabricated metal products. 



63 



ALUMINUM DEMAND-CANS AND CONTAINERS 
THOUSAND SHORT TONS (1960-1976) 



y 

1200 



900 - 



600 - 



300 - 








500 1000 
GNP (BILLION 1972 $> 



1500 



2000 X 



Y == - 1251.4-467 + 
R-SQUARED = 0.9271 



1.7276 X 



PROJECTIONS 
1985 
2000 



Y 
1807.3477 
3572.1366 



X 
17 70.5000 
2792.0000 



FIGURE F-21. - Scatter diagram of aluminum demand for cans and containers use 
and gross national product (GNP). 



:iiS3'!Sic3'T!*iffiSi:!igiiSlS«til5ililii3|fflg»!a8^^^ 



64 



ALUMINUM DEMAND — CANS AND CONTAINERS 
THOUSAND SHORT TONS < 1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 












50 



100 



FRBI-TOTAL PRODUCTION 
1967 = 100 



150 



200 X 



Y = - 852.9654 + 
R~ SQUARED = 0.9089 



13.5006 X 



PROJECTIONS 
1985 
2000 



1662.2002 
3332.2270 



X 
186.3000 
310.0000 



FIGURE F-22. - Scatter diagram of aluminum demand for cans and containers use 
and FRB index for total production. 



''•'•'•''■'■'°~TrTT'' ' ' 'T^ '''''''' '''T '"''''' ^ '' ' '' ' "'"" '' rr ," : ' : r""""! !' ! ' ;,', ' . !"l ' V" ' | 



65 



ALUMINUM DEMAND--CANS AND CONTAINERS 
THOUSAND SHORT TONS (1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 



^ 





50 



100 



150 



200 X 



FRBI-METALWORKING MACHINERY 
1967 = 100 



Y = - 679.7933 + 
R-SQUARED = 0.4-706 



13.9800 X 



PROJECTIONS Y X 

1985 1512.2862 156.8000 

2000 24-88.0971 226.6000 

FIGURE F-23. - Scatter diagram of aluminum demand for cans and containers use 
and FRB index for metalworking machinery. 



M:ig'«R-^«Sgi;SSnSii™l5Si;:;fa:ifMSSf!g|S 



66 



ALUMINUM DEMAND-'-CANS AND CONTAINERS 
THOUSAND SHORT TONS < 1960-1976) 



Y 
1200 



900 - 



600 - 



300 - 












50 



100 



150 



200 X 



FR3I-FABRICATED METAL PRODUCTS 
1967 = 100 



Y = - 1063.7366 + 
R-SQUARED = 0.8778 



15.9112 X 



PROJECTIONS 
1985 
2000 



1747.7850 
3073. 1940 



176.7000 
260 . 0000 



FIGURE F-24. - Scatter diagram of aluminum demand for cans and containers use 
and FRB index for fabricated metal products. 



hii m i M ii i iiii imi iii iM i rt mnii riM il ii lii m iiiiiiii i iii ii li^^ i Miiii iiiiiiiiiiiiiiwtiitninnriiiw«fa**i^^ 



67 



ALUMINUM DEMAND-APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS < 1960-1976) 



V}\6\0 




1 




■ ■ -T 


I 




750 


- 






+ 




^ 


500 














2:50 

i9l 


/ 




/i 




1 





500 1000 
GNP (BILLION 1972 $) 



1500 



2000 X 



Y ^ *- 114.2115 + 
R-SQUARED = 0.7404. 



0.5041 X 



PROJECTIONS Y X 

1985 778.4576 1770.5000 
2000 1293.4882 2792.0000 

FIGURE F-25. - Scatter diagram of aluminum demand for appliances and equipment 
use and gross national product (GNP). 



68 



ALUMINUM DEMAiND-- APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS (1960-1976) 



Y 
1000 



750 



500 



250 - 








75 150 225 300 X 
U.S. POPULATION (MILLION PERSONS) 



Y = - 1111.3208 + 
R-SQUARED = 0.6436 



7.5530 X 



PROJECTIONS 
1985 
2000 



656.8400 
871.3456 



234.1000 
262 . 5000 



FIGURE F-26. - Scatter diagram of aluminum demand for appliances and equipment 
use and U.S. population. 



69 



ALUMINUM DEMAND-~APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS < 1960-1976) 



Y 

1000 



750 



500 - 



250 - 












50 



100 



150 



200 X 



FRBI -TOTAL PRODUCTION 
1967 = 100 



Y = - 17.0310 + 
R-SQUARED = 0.7974- 



4.1297 X 



PROJECTIONS Y X 

1985 752.3395 186.3000 
2000 1 263 . 1 883 310. 0000 

FIGURE F-27. - Scatter diagram of aluminum demand for appliances and equipment 
use and FRB index for total production. 



70 



ALUMINUM DEMAND — APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS (1960-1976) 



Y 
1000 



750 - 



500 - 



250 - 








50 100 150 200 X 

FRBI-MAJOR ELECTRICAL EQUIPMENT AND PARTS 
1967 = 100 



Y = 3.734.7 + 
R-SQUARED = 0.84-91 



4.. 1120 X 



PROJECTIONS 
1985 
2000 



716.3598 
94-5.4.031 



173.3000 
229.0000 



FIGURE F-28. - Scatter diagram of aluminum demand for appliances and equipment use 
and FRB index for major electrical equipment and parts. 



71 



ALUMINUM DEMAND~~APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS < 1960-1976) 



Y 
1000 



750 - 



500 - 



250 












50 



100 



150 



200 X 



FRBI-ELECTRICAL MACHINERY 
1967 = 100 



Y = 93.9233 + 
R-SQUARED = 0.8300 



3.1001 X 



PROJECTIONS Y X 

1985 642.64i6 177.0000 

2000 868 . 949 1 250 . 0000 

FIGURE F-29. - Scatter diagram of aluminum demand for appliances and equipment 
use and FRB index for electrical machinery. 



r!iji5«&ft,iiaai!-£i?Sie«:^*-:;Ssi«)ei^^,w*iia!aE!!!i*!a^^ 



72 



ALUMINUM DEMAND— APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS < 1960-1976) 



Y 

1000 



750 - 



500 - 



250 - 












50 



100 



150 



200 X 



FRBI-HOUSEHOLD APPLIANCES 
1967 = 100 



Y = 58.0345 + 
R-SQUARED = 0.8757 



3.2125 X 



PROJECTIONS 
1985 
2000 



951.434-8 
1631.8454 



X 
278.1000 
489.9000 



FIGURE F-30. - Scatter diagram of aluminum demand for appliances and equipment 
use and FRB index for household appliances. 



73 



ALUMINUM DEMANDS-APPLIANCES AND EQUIPMENT 
THOUSAND SHORT TONS < 1960-1976) 



\o\o\o 








— 1-- — 




/ 


750 


- 






+ 




^ 


500 






+ 
+ / 


+ 






250 


/ 


I 




1 


1 









50 



100 



150 



200 X 



FRBI -COMMUNICATION EQUIPMENT 
1967 = 100 



Y = - 52.8626 + 
R -SQUARED = 0.7009 



4.. 8275 X 



PROJECTIONS 
1985 
2000 



678.9938 
1091.2665 



X 
151.6000 
237 . 0000 



FIGURE F-31. - Scatter diagram of aluminum demand for appliances and equipment 
use and FRB index for communication equipment. 



!:,:i:;ij:;i.-.iv!.t^,iidjB:i;jacm:agnmnmcnairB»iil'mi.nili!nmiiimgM«8iHieiBi!lliW!«H^ 



74 



ALUMINUM DEMAND — MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



Y 

600 



450 - 



300 - 



150 - 








500 1000 
GNP <BILLION 1972 $> 



1500 



2000 X 



Y = - 190.7349 4 
R-SQUARED = 0.8373 



0.4728 X 



PROJECTIONS 
1985 
2000 



Y 
646 . 4556 
1129.4774 



X 
1770.5000 
2792.0000 



FIGURE F-32. - Scatter diagram of aluminum demand for machinery use and gross 
national product (GNP). 



iii it ii iii ii ii ) iii i i iii i ni°»*i'*«i'M^^ ^ 



75 



ALUMINUM DEMAND-- MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



Y 
600 



4-50 



300 - 



150 - 








75 150 225 300 X 
U.S. POPULATION (MILLION PERSONS) 



Y = - 1157.3510 + 
R-SQUARED = 0.7606 



7.2410 X 



PROJECTIONS Y X 

1985 537 . 7728 234. 1000 
2000 743.4179 262.5000 

FIGURE F-33. - Scatter diagram of aluminum demand for machinery use and U.S. 
population. 



«;ssi«^i8SSri:ll:^SgS!tS!l!!Jii!S»!iilii!^ 



76 



ALUMINUM DEMAND-~MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



Y 
600 



4-50 - 



300 



150 - 



^ 





50 



100 



150 



200 X 



FRBI-TOTAL PRODUCTION 
1967 = 100 



Y =^ - 95.8003 + 
R-SQUARED = 0.884-4 



3.8354. X 



PROJECTIONS 
1985 
2000 



618.7365 
1093.1768 



186.3000 
310.0000 



FIGURE F-34. - Scatter diagram of aluminum demand for machinery use and FRB 
index for total production. 



77 



ALUMINUM DEMAND- -MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



DkDi£l 






1 -- 


/ 




450 


- 




4- / 

4- / 
4- / 

4- / 


/ 


- 


300 






4- 






150 


4^ 


i 






- 


f7t 


/ L „ 




1. 











50 



100 



150 



200 X 



FRBI-METALWORKING MACHINERY 
1967 =100 



Y = - 129.1799 + 
R-SQUARED = 0.7098 



4-. 94-45 X 



PROJECTIONS 
1985 
2000 



646.1323 
991 . 2649 



X 
156.8000 
226 . 6000 



FIGURE F-35. - Scatter diagram of aluminum demand for machinery use and FRB 
index for metalworking machinery. 



!lSifliili^S::l^g!||lg8j!!giii§iii!!!l:|i!i|«|^^ 



ji wmif i f »i i w» '!i '' 'i'yyng° rapy 



78 



ALUMINUM DEMAND— MACHINERY 

THOUSAND SHORT TONS < 1960-1976) 



Y 
600 



4-50 - 



300 - 



150 












50 



100 



150 



200 X 



FRBI-SPECIAL AND GENERAL INDUSTRIAL EQUIPMENT 
1967 = 100 



Y - - 137.3615 + 
R~SQUARED = 0.8601 



4.6364 X 



PROJECTIONS 
1985 
2000 



769.9851 
1318.0095 



X 
195.7000 
313.9000 



FIGURE F-36. - Scatter diagram of aluminum demand for machinery use and FRB 
index for special and general industrial equipment. 



79 



ALUMINUM DEMAND — DISSIPATIVE 

THOUSAND SHORT TONS < 1960-1976) 



Y 
600 



450 



300 - 



150 - 











■ 1 


1 y 




- 




4f y 

4- 44 




- 


- 




7 4- 

y 

r/ 4- 




- 


- 


/ 


/4f- 




- 




/ 










/ L , 


L 


i 





500 1000 
GNP (BILLION 1972 $> 



1500 



2000 X 



Y = - 114-. 0878 + 
R-SQUARED = 0.624.9 



0.4.106 X 



PROJECTIONS 
1985 
2000 



612.984.9 
1032.4737 



X 
1770.5000 
2792.0000 



FIGURE F-37. - Scatter diagram of aluminum demand for dissipative use and gross 
national product (GNP). 



^aiiii^sisiiiiili^'iliiiifeiiBiii^ 



80 



ALUMINUM DEMAND— DISSIPATIVE 

THOUSAND SHORT TONS (1960-1976) 



0161161 




1 


1 / 




4-50 


- 






•• 


300 






7 
A 




150 


1 


/ 


1 





75 150 225 300 X 
U.S. POPULATION (MILLION PERSONS) 



Y = ~ 927.4-367 + 
R-SQUARED = 0.54-4.3 



6.1579 X 



PROJECTIONS Y X 

1985 514.134-9 234-. 1000 
2000 689.0202 262.5000 

FIGURE F-38. - Scatter diagram of aluminum demand for dissipative use and U.S. 
population. 



81 



ALUMINUM DEMAND — DISSIPATIVE 

THOUSAND SHORT TONS (1960-1976) 



Y 
600 



450 - 



300 - 



150 - 



^ 





50 



100 



150 



200 X 



FRBI-TOTAL PRODUCTION 
1967 = 100 



Y = - 37.5917 + 
R-SQUARED = 0.6836 



3.3900 X 



PROJECTIONS 
1985 
2000 



593-9696 
1013.3155 



X 
186.3000 
310.0000 



FIGURE F-39. - Scatter diagram of aluminum demand for dissipative use and FRB 
index for total production. 



82 



ALUMINUM DEMAND--DISSIPATIVE 

THOUSAND SHORT TONS < 1960-1976) 



Y 
600 



450 - 



300 - 



150 - 












50 



100 



150 



200 X 



FRBI-CHEMICALS AND PRODUCTS 
1967 = 100 



Y = 109.1964 + 
R-SQUARED = 0.5267 



1.7753 X 



PROJECTIONS 
1985 
2000 



Y 
628.4770 
1248.9507 



292 . 5000 
642 . 0000 



FIGURE F-40. - Scatter diagram of aluminum demand for dissipative use and FRB 
index for chemicals and products. 



83 



ALUMINUM DEMAND~-DISSIPATIVE 

THOUSAND SHORT TONS < 1960-1976) 



Y 

600 



4-50 - 



300 - 



150 












50 



100 



150 



FRBI-BASIC CHEMICALS 
1967 = 100 



200 X 



Y = 71.2801 4- 
R-SQUARED = 0.5572 



2.1378 X 



PROJECTIONS Y X 

1985 537.9639 218.3000 
2000 787 . 4-463 335 . 0000 

FIGURE F-41. - Scatter diagram of aluminum demand for dissipative use and FRB 
index for basic chemicals. 



84 



ALUMINUM DEMAND--OTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 
400 



300 - 



200 - 



100 - 








500 1000 
GNP (BILLION 1972 $) 



1500 



2000 X 



Y - 214-.6663 + - 
R~SQUARED = 0.1135 



0.1114 X 



PROJECTIONS Y X 

1985 17.4239 17 70.5000 

2000 - 96.3762 2792.0000 

FIGURE F-42. - Scatter diagram of aluminum demand for other metal use and gross 
national product (GNP). 



85 



(ALUMINUM DEMAND-MOTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 
400 



300 



200 - 



100 - 








75 150 225 
U.S. POPULATION (MILLION PERSONS) 



300 X 



Y = 479.9729 + - 
R-SQUARED = 0.1271 



1.8938 X 



PROJECTIONS 
1985 
2000 



Y 

36.6120 
17. 1746 



X 
234. 1000 
262 . 5000 



FIGURE F-43. - Scatter diagram of aluminum demand for other metal use and U.S. 
population. 



86 



ALUMINUM DEMAND—OTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 
400 



300 - 



?.00 - 



100 












75 



150 



225 



300 X 



GROSS PRIVATE DOMESTIC INVESTMENT 
BILLION 1973 $ 



Y = 14-8.954-6 + - 
R-SQUARED = 0.0231 



0.2925 X 



PROJECTIONS Y X 

1985 57.1052 314-. 0000 

2000 - 3 . 1 526 520 . 0000 

FIGURE F-44. - Scatter diagram of aluminum demand for other metal use and gross 
private domestic investment. 



87 



ALUMINUM DEMAND--OTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



V 
400 



300 - 



200 - 



100 - 












50 



100 



150 



200 X 



FRBI -TOTAL PRODUCTION 
1967 = 100 



Y = 179.8416 4 - 
R-SQUARED = 0.0893 



0.7799 X 



PROJECTIONS 
1985 
2000 



Y 
34.5418 
61.9347 



X 
186.3000 
310.0000 



FIGURE F-45. - Scatter diagram of aluminum demand for other metal use and FRB 
index for total production. 



88 



ALUMINUM DEMAND — OTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 

4.00 



300 - 



200 - 



100 - 



^ 





50 



100 



150 



F RBI -PA I NTS 
1967 = 100 



200 X 



Y = 177.1799 + - 
R-SQUARED = 0.0516 



0.7544 X 



PROJECTIONS 
1985 
2000 



25.0798 
58 . 5903 



X 
201.6000 
312.5000 



FIGURE F-46. - Scatter diagram of aluminum demand for other metal use and FRB 
index for paints. 



89 



ALUMINUM DEMAND~-OTHER METAL 

THOUSAND SHORT TONS < 1960-1976) 



Y 
400 



300 



200 - 



100 












50 



100 



150 



200 X 



FRBI-BASIC CHEMICALS 
1967 = 100 



Y = 183.7706 + - 
R-SQUARED =? 0.1731 



0.7582 X 



PROJECTIONS 
1985 
2000 



Y 
18.2511 
70.2331 



X 



218.3000 
335.0000 



FIGURE F-47. - Scatter diagram of aluminum demand for other metal use and FRB 
index for basic chemicals. 



90 



ALUMINUM DEMAND — REFRACTORIES 

THOUSAND SHORT TONS (1960- 1976) 



Y 
300 



225 



150 - 



75 - 








500 1000 
GNP (BILLION 1972 $> 



1500 



2000 X 



Y = ^ 24-1.7664 + 
R-SQUARED = 0.6014 



0.3685 X 



PROJECTIONS Y X 

1985 410.6825 1770.5000 
2000 787.1167 2792.0000 

FIGURE F-48. - Scatter diagram of aluminum demand for refractories use and gross 
national product (GNP). 



91 



ALUMiriUM DEMANDS-REFRACTORIES 

THOUSAND SHORT TONS < 1960-1978) 



Y 



vljuw 


I ■-' T" - ■ 


1 




4- / 
4- / 




?25 


** 




/ 


" 


150 


- 




1± 


- 








L 




75 


- 




/ 


- 


^ 


L... 




' L 





75 150 225 300 
U.S. POPULATION (MILLION PERSONS) 



X 



Y = - 879.8362 + 
R- SQUARED ^ 0.4-693 



5.0966 X 



PROJECTIONS Y X 

1985 313.2998 234.1000 
2000 4-58 . 0459 262 . 5000 

FIGURE F-49. - Scatter diagram of aluminum demand for refractories use and U.S. 
population. 



92 



ALUMINUH DEMAND--REFRACTORIES 

THOUSAND SHORT TONS < 1960-1976) 



Y 

300 



225 - 



150 



75 












75 



150 



225 



300 X 



GROSS PRIVATE DOMESTIC INVESTMENT 
BILLION 1973 $ 



Y - - 95.4121 + 
R-SQUARED ^ 0.424-1 



1.4924 X 



PROJECTIONS 
1985 
2000 



373.2155 

680.8591 



X 
314.0000 
520 . 0000 



FIGURE F-50. - Scatter diagram of aluminum demand for refractories use and gross 
private domestic investment. 



93 



ALUMINUM DEMAND—REFRACTORIES 

THOUSAND SHORT TONS (1960-1976) 



Y 

300 



225 - 



150 - 



75 - 












50 



100 



FRBI -TOTAL PRODUCTION 
1967 =100 



150 



200 X 



Y = - 194-. 3504 + 
R-SQUARED = 0.6840 



3.2087 X 



PROJECTIONS 
1985 
2000 



403.44.54. 
800.3716 



X 
186.3000 
310.0000 



FIGURE F-51. - Scatter diagram of aluminum demand for refractories use and FRB 
index for total production. 



94 



ALUMINUM DEMANDS-REFRACTORIES 

THOUSAND SHORT TONS < 1966-1976) 



Y 
300 



2?5 - 



150 



75 - 



A. 





50 



100 



FR3I-IR0N AND STEEL 
1967 ^ 100 



150 



200 X 



Y = - 169.4929 + 
R-SQUARED = 0.3764 



3.2071 X 



PROJECTIONS 
1985 
2000 



Y 
332.4-187 
484.7561 



X 
156.5000 
204 . 0000 



FIGURE F-52. - Scatter diagram of aluminum demand for refractories use and FRB 
index for iron and steel. 



95 



ALU ^>INUi DE.iAMD — REFRACTORIES 

IHO'JSAND SHORT TOf-^S (1666- 1876) 



^yju^ 




■T 


-1 

4 / 
4- / 


/ 




2?5 






4 + / 






150 


" 








- 


75 


- 




/ 




- 


Oi 




I / 


L 


k 









50 



100 



150 



200 X 



FR3I-BASIC STEEL AND MILL PRODUCTS 
1967 ^ 100 



Y = - 2 75.5787 4 
R-SQUARED -•■ 0.4.083 



3.2483 X 



PROJECTIONS y X 

1985 337 . 0083 157. 8000 
2000 509.4-94.2 210.S000 

FIGURE F-53. - Scatter diagram of aluminum demand for refractories use and FRB 
index for basic steel and mill products. 



96 



Y 
400 



300 - 



200 - 



100 - 







ALUMINUM DEMAND--CHEMICALS 
THOUSAND SHORT TONS <1960-"ig76> 




500 1000 
GNP (BILLION 1972 $> 



1500 



2000 X 



Y = - 4-82.2833 + 
R-SQUAKED = 0.7079 



0.6831 X 



PROJECTIONS Y 

1985 727.24-60 

2000 14-25.0908 



1770.5000 
2792.0000 



FIGURE F-54. - Scatter diagram of aluminum demand for chemicals use and gross 
national product (GNP). 



97 



ALUMINUM DEMAND'--CN£MICALS 

THOUSAND SHORT TONS < 1960-1976) 



^v>}(J 




1 


) 


1 1 
41 








~ 




r 


• 


300 








J"^ 




200 


- 






/ 


- 


100 


- 






f 


- 


n 




L 


L 


/ t 





75 150 225 300 X 
U.S. POPULATION (MILLION PERSONS) 



Y = - 1938.4.867 + 
R-SQUARED ^ 0.7181 



10.7716 X 



PROJECTIOMS Y X 

1 985 S83 . 1 656 234 . 1 000 
2000 889.0815 262.5000 

FIGURE F-55. - Scatter diagram of aluminum demand for chemicals use and U.S. 
population. 



98 



ALUMINUM DEMAND — CHEMICALS 

THOUSAND SHORT TONS (1960-1976) 



Y 
400 



300 



200 - 



100 



^ 





7b 



150 



GROSS PRIVATE DOMESTIC 
BILLION 1973 $ 



2?5 

INVESTMENT 



300 X 



Y = - 119.1208 -»- 
R-SQUARED ~ 0.3319 



2.2558 X 



PROJECTIONS 
1985 
2000 



589 . 2236 
1053.9337 



X 
31 4.. 0000 
520.0000 



FIGURE F-56. - Scatter diagram of aluminum demand for chemicals use and gross 
private domestic investment. 



99 



ALUMINUM DEMANDS-CHEMICALS 

THOUSAND SHORT TONS < 1960-1976) 



Y 
400 



300 



200 - 



100 - 












50 



100 



150 



200 X 



FR8I-T0TAL PRODUCTION 
1967 = 100 



Y = - 335.8693 + 
R-SQUARED = 0.6727 



5.4373 X 



PROJECTIONS Y X 

1 985 677. 0999 1 86 . 3000 
2000 1349.6942 310.0000 

FIGURE F-57. - Scatter diagram of aluminum demand for chemicals use and FRB 
index for total production. 



100 



ALUMINUM DEMAND— CHEMICALS 

THOUSAND SHORT TONS (1960-1976) 



y 

400 



300 - 



200 - 



100 - 












50 



100 



FRBI-BASIC CHEMICALS 
1967 = 100 



150 



200 X 



Y = - 118.8334 + 
R -SQUARED = 0.7406 



3.1873 X 



PROJECTIONS 
1985 
2000 



576.9683 
948.9339 



218.3000 
335.0000 



FIGURE F-58. - Scatter diagram of aluminum demand for chemicals use and FRB 
index for basic chemicals. 



101 



ALUMINUM DEMAND — CHEMICALS 

THOUSAND SHORT TONS <19B0-197B) 



400 



300 " 



200 - 



100 - 












50 



100 



150 



FRBI-CHEMICALS AND PRODUCTS 
1967 = 100 



200 X 



Y = - 58.724-2 + 
R-SQUARED = 0.7185 



2.6294 X 



PROJECTIONS 
1985 
2000 



710.3981 
1629.4006 



X 
292.5000 
642 . 0000 



FIGURE F-59. - Scatter diagram of aluminum demand for chemicals use and FRB 
index for chemicals and products. 



102 



ALUMINUM DEMAND--CHEMICALS 

THOUSAND SHORT TONS < 1960-1976) 
Y 
400 r 



300 - 



2:00 - 



100 












73 



150 



225 



300 X 



FRBI -PLASTICS PRODUCTS. NEC. 
1967 = 100 



y = 95.0251 + 
R-SQUARED = 0.7816 



1.0389 X 



PROJECTIONS Y X 

1985 577.324.8 464.2000 
2000 1 024 . 2989 884 . 4000 

FIGURE F-60. - Scatter diagram of aluminum demand for chemicals use and FRB 
index for plastics products, n.e.c. 



103 



ALUMIMUM DEMAND — CHEMICALS 

THOUSAND SHORT TONS < 1966-1976) 



Y 

40D 



300 



200 



100 












FR8I-TEXTI 
1967 = 100 



50 100 150 
E MILL PRODUCTS 



200 X 



Y 



- 24-5.0129 -f 
R-SQUARED ^ 0.7^4-8 



4.4-497 >: 



PROJECTIONS 
198S 
2000 



Y 
614.2427 
1103.2753 



X 
193.1000 
303 . 0000 



FIGURE F-61. - Scatter diagram of aluminum demand for chemicals use and FRB 
index for textile mill products. 



104 



ALUMINUM DEMAND — ABRASIVES 

THOUSAND SHORT TONS < 1960-1976) 



iZVJiO 










/ 


150 


- 




+ 
+ / 




- 


100 


- 




A 

A- 
+ /+ 

r 




- 


50 


- 




/ 




- 


f^ 




,/ 


1 


1. 





500 1000 
GNP (BILLION 1972 $> 



1500 



2000 X 



Y = - 97.1890 + 
R-SQUARED = 0.4932 



0.1679 X 



PROJECTIONS 
1985 
2000 



Y 
200.194-2 
371.7711 



X 
1770.5000 
2792.0000 



FIGURE F-62. - Scatter diagram of aluminum demand for abrasives use and gross 
national product (GNP). 



105 



ALUMINUM DEMANDS-ABRASIVES 

THOUSAND SHORT TONS < 1960- 1978) 



Y 
200 



150 - 



100 ~ 



50 - 








75 150 2ZS 300 X 

U.S. POPULATION (MILLION PERSONS) 



Y =^ - 359.5734- + 
R-SQUARED - 0.34-06 



2.1853 X 



PROJECTIONS 
1985 

2000 



Y 
152.014-3 
214-.0779 



X 
234-. 1000 
262 . 5000 



FIGURE F-63. - Scatter diagram of aluminum demand for abrasives use and U.S. 
population. 



106 



ALUMINUM DEMAND-~ABRASIVES 

THOUSAND SHORT TONS < 1980-1976) 



200 



150 - 



100 - 



50 - 












75 



150 



225 



300 X 



GROSS PRIVATE DOMESTIC INVESTMENT 
BILLION 1973 $ 



Y ~ - 57.7339 + 
R-SQUARED = 0.5201 



0.8318 X 



PROJECTIONS Y X 

1985 203.^614 31^.0000 
2000 37 4.8189 520.0000 

FIGURE F-64. - Scatter diagram of aluminum demand for abrasives use and gross 
private domestic investment. 



107 



ALUMINUM DEMAND--ABRASIVES 

THOUSAND SHORT TONS <1966~1976> 
Y 
200 r 



150 ~ 



100 



50 - 












50 



100 



FR3I~IR0N AND STEEL 
1967 ~ 100 



150 



200 X 



Y = ~ 120.3796 + 
R-SQUARED = 0.5707 



1.9875 X 



PROJECTIONS 
1985 

2000 



190.6716 
285.0801 



X 
158.5000 
204.. 0000 



FIGURE F-65. - Scatter diagram of aluminum demand for abrasives use and FRB 
index for iron and steel. 



108 



ALUMINUM DEMAND--ABRASIVES 

THOUSAND SHORT TONS (1966-1976) 



Y 



diOiC^ 




1 


1 


T / 




150 


- 




+ / 


/ 


- 


100 


~ 




V 




- 


50 


-■ 


/ 


^ 




"- 


m 




/ I 




1 









50 



100 



150 



200 X 



FRBI-METALWORKING MACHINERY 
1967 = 100 



Y = - 43.324-4 4 
R-SQUARED - 0.307 7 



1 .4448 X 



PROJECTIONS 
1985 
2000 



Y 
183.2223 

284.0703 



X 
156.8000 
226.6000 



FIGURE F-66. - Scatter diagram of aluminum demand for abrasives use and FRB 
index for metalworking machinery. 



109 



ALUMINUM DEMAND — ABRASIVES 

THOUSAND SHORT TONS (1966-1S/6) 



Y 
200 



150 



100 



50 - 












50 



FRBI-8ASIC STEEL 
1967 = 100 



100 150 200 
AND MILL PRODUCTS 



X 



Y = - 122.8361 + 
R-SQUARED ~ 0.6114 



2.0008 X 



PROJECTIONS Y X 

192.8972 157.8000 
299.14^21 210.9000 

Scatter diagram of aluminum demand for abrasives use and FRB 
index for basic steel and mill products. 

ft U. S. GOVERNMENT PRINTING OFFICE : 



1985 
2000 

FIGURE F-67. 



1980—327-748 



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